UC-NRLF 


SB    EflD    mi 


THE  TRANSITION  CURVE 


OR 


CURVE  OF  ADJUSTMENT 

AS   APPLIED   TO   THE   ALIGNMENT   OF 
RAILROADS 


BY  THE  METHOD  OF  RECTANGULAR 

CO-ORDINATES    AND    BY 

DEFLECTION  ANGLES 

(OR  POLAR  CO-ORDINATES) 


BY 

N.    B.    KELLOGG 

M.  AM.  SOC.  C.  E. 


THIRD    EDITION 


NEW   YORK 

McGRAW  PUBLISHING  COMPANY 
1907 


COPYRIGHTED,  1899,  1904, 

BY 
N.  B.  KELLOGG 


Stanhope  press 

F.    H.     GILSON     COMPANY 
BOSTON     U.  S.  A. 


PKEFACE 


ERRATA 

Page  8,  Equation  (17),  read:   ^p  when-  =  0. 

Page  22,  Line  5,  read:  With  instrument  at  A,  PA  -  0, 

since  D  =  0. 
Page  23,  Second  line  above  footnote:  change  factor  to 

term. 
Page  37,  Last  line  of  Problem  III,  Case  2,  read:  as  in 

the  first  part  of  this  problem. 
Page  46,  Table,  5th  column,  near  bottom:  change  7.35  to 

6.62. 
Page  46,  Table,  10th  column,  middle:  change  1.07  to 

1.57. 

Page  65,  Table,  7th  column,  top:  change  08.5'  to  07.5'. 
Page  69,  near  bottom  of  page,  read:  External  Sec. a  = 

Sec.a-L 


Nordling  equations,  the  formulae  (42-46)  were  obtained 
by  the  writer  during  the  summer  of  1884,  as  were  also 
§  (2—9)  inclusive,  by  an  independent  process,  resulting 
in  the  verification  of  the  formulae  invented  by  Froude  in 
1841  and  published  in  1861.  Hence,  it  is  thought  that 

366471 


Stanbopc  press 

F.    H.    GILSON     COMP 
BOSTON     U.S.A. 


PREFACE 


WHILE  the  theory  of  the  transition  curve  as  here  devel- 
oped is  based  upon  the  methods  given  by  M.  Nordling 
(see  "Annales  des  Fonts  et  Chausse*es,"  1867),  the  equa- 
tions for  both  the  cubic  parabola  and  the  spiral  appear 
in  Professor  Rankine's  "Civil  Engineering"  (Ed.  1863), 
and  are  accredited  to  Dr.  William  Froude,  undoubtedly 
ante-dating  those  of  Nordling. 

Translations  of  a  portion  of  Nordling's  demonstration 
have  appeared  from  time  to  time,  but  only  so  far  as 
related  to  connecting  a  straight  line  with  a  circular 
curve.  That  portion  relating  to  connecting  circular 
curves  of  different  radii,  by  means  of  the  cubic  parabola, 
'has  not  appeared  in  the  form  given  by  Nordling,  so  far 
as  I  am  aware.  The  formulae  deduced  in  the  latter  case 
are  of  general  application  and  equally  true  for  connecting 
curve  with  curve,  or  curve  with  tangent,  when  proper 
values  are  introduced  into  the  equations. 

That  of  joining  a  tangent  with  a  circular  curve,  by 
means  of  the  transition  curve,  is  a  special  case  where  one 
of  the  radii  becomes  infinitely  great.  Some  of  the  recent 
spirals,  adopted  as  curves  of  adjustment  in  railroad 
location,  easily  develop  from  the  equations  of  the  cubic 
parabola  by  making  the  proper  substitutions  in  them. 

Following  the  supposition  indicated  by  M.  Nordling, 
i.e.,  regarding  x  =  L  and  substituting  L  for  x  in  the 
Nordling  equations,  the  formulae  (42-46)  were  obtained 
by  the  writer  during  the  summer  of  1884,  as  were  also 
§  (2-9)  inclusive,  by  an  independent  process,  resulting 
in  the  verification  of  the  formulae  invented  by  Froude  in 
1841  and  published  in  1861.  Hence,  it  is  thought  that 

360471 


IV  PREFACE 

the  curve  herein  presented  may,  not  improperly,  be 
called  Froude's  Spiral.  Since  the  spiral,  as  an  adjust- 
ment curve,  is  free  from  some  defects  noticeable  in  the 
cubic  parabola  (appearing  in  the  edition  of  1899),  the 
latter  is  omitted  from  the  present  edition. 

The  Compound  Circular  Transition  Curve,*  known  also 
3,s  the  " Railroad  Spiral"  (an  approximation  to  the  Froude 
Spiral),  though  giving  good  results  as  an  adjustment 
curve,  is  defective  in  not  being  expressed  by  a  formula 
following  a  law  of  uniform  change  of  curvature  for  con- 
secutive points  throughout  its  entire  length.  A  chapter 
on  the  compound  circular  curve  was  prepared  simul- 
taneously with  that  of  the  true  spiral  and  the  cubic 
parabola,  but  was  abandoned  for  the  reason  that  the 
deduced  formulae  were  found  to  be  not  easily  transposed 
or  modified  for  arbitrary  values  of  the  offset  distance 
and  for  the  computation  of  fractional  chords. 

Few  engineers,  at  the  present  time,  will  locate  an 
important  railway  line  without  the  use  of  some  good 
curve  of  adjustment. 

A  few  simple  methods  of  the  calculus  are  used  to 
derive  the  formulae,  but  a  knowledge  of  only  the  ordinary 
processes  of  algebra  and  trigonometry  is  required  for 

*  If  a  compound  circular  transition  curve  and  a  spiral  be  run  for 
the  same  offset,  the  origin  of  the  compound  circular  curve  will  be 
fonnd  one-half  chord  length  in  advance  of  that  of  the  true  spiral 
and  the  terminus  of  the  compound  circular  curve  one-half  chord 
length  back  of  that  of  the  spiral,  nearly,  i.e.,  the  ends  of  the  chorda 
of  the  spiral  correspond  closely  to  the  middle  points  of  the  several 
arcs  of  the  compound  circular  transition  curve.  For  instance,  a 
compound  circular  curve  of  nine  equal  chords  will  have,  nearly, 
the  same  offset  (/)  as  a  spiral  of  ten  equal  chords  of  the  same 
length,  each  curve  terminating  in  the  same  radius  of  curvature  as 
the  main  curve;  the  total  curvatures,  however,  will  differ,  since  the 
spiral  is  tangent  to  the  main  curve  nearly  one-half  chord  length 
beyond  where  the  compound  circular  curve  would  become  tangent 
to  the  main  curve. 


PREFACE  V 

their  application.     It  is   seldom  necessary  to  use  more 
than  two  terms  in  the  formulae  involving  series. 

The  larger  type  may  be  read  independently  of  the 
smaller  type.  The  latter  may  be  used  when  the  curve 
is  extended  beyond  ordinary  limits  and  greater  accuracy 
in  results  is  sought. 

For  field  practice,  formulae  (42-52),  (60-61),  are  the 
ones  with  which,  taken  with  the  tables,  one  should 
become  familiar ;  and  these  formulae  may  be  regarded  as  a 
summary  of  the  essential  part  of  the  book.* 

The  "run  off"  is  understood  to  be  equal  in  length  to 
that  of  the  transition  curve  and  coincident  with  it. 

The  transition  or  adjustment  curve  can  rarely  be 
applied  to  switches  or  turnouts  to  advantage,  except 
where  high  speeds  are  maintained. 

The  more  important  formulae  are  in  full-faced  type. 

I  am  indebted  to  Professor  H.  I.  Randall,  C.E.,  of  the 
University  of  California,  and  also  to  Mr.  D.  E.  Hughes, 
C.E.,  for  valuable  suggestions.  The  latter  is  the  author 
of  an  excellent  paper  on  this  subject. 

N.  B.  K. 

SAN  FRANCISCO,  January,  1904. 

*  By  making  all  terms  containing  the  factors  m  or  /,  representing 
spiral  offsets  in  Probs.  (I-V)  equal  zero,  the  spiral  disappears  from 
the  formula  and  it  stands  for  circular  curves  simple  or  compound  as 
the  case  may  be. 

Except  in  a  few  cases,  Greek  letters  have  been  used  to  designate 
angles,  small  Roman  italics  for  lines  and  Roman  capitals  for  points. 

S.  F.  1907. 


CONTENTS 


THE   SPIRAL 

SECTION.  PAGE. 

1     DEFINITION   AND   OBJECT   OF   THE   TRANSITION 

CURVE     ....    ..,„..,   '^v* 1 

1  FUNDAMENTAL  EQUATION  OF  THE  CURVE   ...  3 

2  EQUATION  FOR  COMPOUND  CURVES 3 

3  CENTRAL  ANGLE  FOR  COMPOUND  CURVES   ...  5 

4  DIFFERENCE  IN  LENGTH  OF  SPIRAL  AND  CIRCU- 

LAR  ARCS   SUBTENDING   SAME   ANGLE    ....  6 

5  TRANSFORMATION    TO    RECTANGULAR    CO-ORDI- 

NATES         7 

6  RECTA-NGULAR    CO-ORDINATES    AT    THE    OFFSET 

DISTANCE 11 

7  OFFSET  DISTANCE.      DISTANCE  BETWEEN  CEN- 

TERS         ....  13 

8  PRACTICAL  FORMULAE 15 

9  METHOD  BY  DEFLECTION  ANGLES 18 

10     ORDINATES  FROM  LONG  CHORD 24 

10     REMARKS    ON    SUPERELEVATION,    ETC.     TABLE 

OF  RADII  AND  THEIR  RECIPROCALS 25 


PROBLEMS 

11  PROBLEM  I.     SEMI-TANGENTS    .    .___.    .  ^  ...  27 

12  EXTERNAL  SECANTS      29 

13  PROBLEM  II.     LOCATION  OF  OFFSET  "/"  ...  31 

14  PROBLEM  III.     COMPOUND  CURVES 33 

15  PROBLEM  IV.     TANGENTS  TO  Two  CURVES   .    .  37 

16  TRANSITION  CURVE  IN  OLD  TRACK 40 

vii 


Vlll  CONTENTS 

SECTION.  PAGE. 

17  EXPLANATION    OF    TABLES.     LAYING    OUT    BY 

RECTANGULAR  CO-ORDINATES 42 

18  LAYING  OUT  BY  DEFLECTION  ANGLES     ....     43 

TABLES 

LENGTH  OF  CIRCULAR  ARCS  AT  RADIUS  =  1  .    .     45 
MINUTES  IN  DECIMAL  OF  DEGREE    ......     46 

TRANSITION  CURVE  TABLES 46 

APPENDIX 
MISCELLANEOUS  PROBLEMS  AND  TABLES     ...     59 


THE    TRANSITION   CURVE 


§  i.  The  true  transition  curve  is  one  of  which  the 
radius  of  curvature,  at  its  origin,  is  infinitely  great; 
and  at  any  other  of  its  points,  the  radius  of  curvature  is 
inversely  proportional  to  the  distance  of  the  point, 
measured  on  the  curve,  from  the  origin;  the  product  of 
the  radius  and  distance  being  a  constant  for  all  points 
of  the  curve. 

The  object  of  introducing  the  transition  curve  between 
circular  curves  of  different  radii,  or  between  a  tangent 
and  circular  curve,  as  applied  to  the  alignment  of  rail- 
roads, is  to  give  centrifugal  force  an  appreciable  time  to 
develop,  from  that  due  to  one  given  radius  of  curvature 
to  that  of  another,  in  a  moving  body  passing  from  a 
circular  path  of  one  rate  of  curvature  to  that  of  another 
rate  of  curvature;  and  to  develop  simultaneously  a  force 
equal  and  opposed  to  the  centrifugal  force,  neutralizing 
it  at  every  point  of  the  curve.  If  one  of  the  given  radii 
is  made  infinitely  great,  its  curve  becomes  a  tangent; 
and  centrifugal  force  develops  gradually  from  zero  to 
that  due  to  the  other  given  radius  of  curvature,  in  the 
time  it  takes  to  traverse  the  transition  curve;  thus 
avoiding  instantaneous  development  or  "shock." 

The  opposing  force  is  the  horizontal  component  of  a 
force  due  to  gravity  developed  by  inclining  the  vertical 
axis,  passing  through  the  center  of  gravity  of  the  moving 
body,  from  a  normal  to  the  plane  of  rotation  and  towards 
the  center  of  rotation. 

From  mechanics  the  expression  for  centrifugal  force 
1 


SITION*   CURVE 


/  of  a  weight  -io  moving  iu  a  circular  path  with  a  radius  r 
and  a  velocity  v  is  : 

,  _     wv2 
~  ' 


The    opposing    horizontal    force    due    to     gravity    is 
(Fig.  1): 


we 

w,  =  —  , 

g 


in  which  w  is  the  weight  of  the  moving  body,  g  the  width 
of  the  path  or  gauge  of  the  track,  and  e  the  inclination 
of  the  path,  or  "cant,"  towards  the  center  of  rotation, 
in  a  distance  g. 

Since    by   the  hypothesis    the    opposing   forces    equal 
each  other  in  intensity, 


solving  for  e 


nearly.* 


*  More  correctly  (Fig.  1)  e 


hv2 


V  (32.2)2  r2  +  v* 

in  which  h  —  gauge  of  track. 

The  resultant  of  w  and  w/  due  to 
the  effect  of  gravity  =  vV  +  Wli 
which  is  the  pressure  normal  to  the 
plane  of  the  path  when  centrifugal 
force  is  developed. 

It  may  not  be  out  of  place  to 
remark  here,  that  it  is  the  usual 
custom  in  fixing  superelevation,  to 
depress  the  inner  rail  below  the 
grade  of  the  center  line  of  the  track 

%e  and  elevate  the  outer  rail  above  the  grade  line  %e,  thus  main- 
taining the  center  line  at  grade. 


Fig.  I. 


THE    SPIRAL  3 

If  i  denote  the  distance,  measured  on  the  curve,  re- 
quired to  change  the  inclination  or  "cant"  of  the  path 

one  unit,  -  will  denote  the  change  of  cant  in  one  unit  of 

distance.     Therefore,  in  a  distance  OB4  =  Z/y  the  cant,  or 
superelevation,  will  be: 


i  32.27-, 

from  which 


The  second  member  of  Eq.  (2)  is  a  constant,  since 
all  its  factors  are  assumed  constant,  and  represented  by 
P,  whence,  the  fundamental  equation  of  the  curve: 

If,  '=  P,  (3)* 

which  is  the  equation  of  a  spiral,  in  which  OB,  =  L/} 
Fig.  2,  represents  the  required  length  of  the  transition 
curve,  DB,  =  r,  the  radius  of  the  curvature  common  to 
the  transition  curve  and  the  given  circular  curve  where 
these  curves  become  tangent  each  other.  (See  Fig.  2.) 


GENERAL    FORMULA. 

§  2.  If  h?  the  figure  we  let  OB,  be  denoted  by  Lt  and 
OBn  by  Ly/,  then 

OB,  =£,  =f;  OBit  =  Lti  =  f; 

Ti  '// 

subtracting,  letting    Lu  —  L,  =  L,  we  have  the  General 
Formula  for  compound  curves. 

*  Froude's  Eq.  (3),  is  in  the  form  of  L  =  ei,  which  is  called  by 
Rankine  "The  Curve  of  Adjustment."  Any  particular  spiral  is 
designated  by  the  numerical  value  assigned  to  its  constant  P. 


TRANSITION    CURVE 


which  is  the  ideal  equation  of  all  Transition  Curves, 
Bt  being  the  first  and  Blt  the  second  point  of  compound. 


.A......^ 

^V- 


E 
M 


r-TT^v^S 

\__^vy:i:z::Wa    c 

.*>.  A     <.  /  /n     r  >>.. 


Tpv^ 


•• «'    =*         ,-^v  '2r 

-t^  * 


It  fulfills  every  condition  of  theory  and  offers  no  unusual 
difficulties  in  application. 

When  the  length  L  of  the  spiral  and  the  radii  r,,  and 
ry  are  assumed. 

The  constant  P  =  (  Lr//r/  } 
V,  -  rj 

L 


or 


(5) 


THE    SPIRAL  5 

•p 

§  3.   In  general  let  L  =  ~  =  Pr'1  ;  differentiating  dL  =  Pr~zdr. 

We  have  from  the  calculus:  dL/  =  rda,  whence 


rda  =  Pr-*dr,  da  =  Pr~*dr'  =-  Pr^dr\  or 


hence  for  any  two  arcs  atj  and  a,  at  radius  =  1,  we  have: 
P  P 

a"  =  2^'a'==2^; 
p       p      PI  1      1 


P/l        1\    /I        1  \ 

a//  -  a   =FT   ---        --  1  --   * 
2  lr^       rj    \r»       rj 


substituting  in  value  for  att  —  a,,  we  have  for  the  Central 
Angle  for  Compound  Curves  expressed  in  arc  at  radius 
=  1. 


*  The  values  of  £/  &,/  a.,  a.//  and  <£  are  expressed  in  arc  at 
radius  =  1.  The  degrees  in  an  arc  of  a  circle  which  is  equal  to  the 
radius  in  length  =  57.295.  The  arc  of  1  degree  at  radius  =  1  is 
.01745  -f .  Hence  the  expression  for  any  number  of  degrees  in  a 
given  circular  arc  is:  A°  =  ^745  +  * 


6  TRANSITION    CURVE 

in  which  0  =  ft/  ±  ft  (9) 

is  the  angle  subtended  by  the  arc  L  of  the  spiral  as  well  as 
the  sum  of  the  circular  arcs  A4B,  +  AtlBn 

the  +  sign  being  used  when  the  directions  of  curvature  are 

similar; 
the  —  sign  when  the  directions  of  curvature  are  reversed. 

EXAMPLE  No.  1.   GivenL  =  150;  rn  =  818.8;  r  =  2865; 

to  find  arc     (a,,  -  a,)  =  0  =  ft,  +  ft  =  £  (-L  +  i)  , 

~  w/       '// 

arc  („„  -  n/)  .  0  .  if°  (gjj-g  +  ;^-5)  -  .11775 

=  arc  of  6°  45'. 

If  in  equation  (8)  we  make  rt  =  oo  ,  and  ax  =  0,  then 
—  - 


the  angle,  expressed  in  arc,  subtended  by  spiral  OB/X. 

§  4.  DIFFERENCE  IN  LENGTH  OF  THE  SPIRAL  AND  THE 
SUM  OF  THE  CIRCULAR  ARCS  SUBTENDING  THE  SAME  ANGLE. 

The  rectangular  co-ordinates  of  the  center  of  the  circle  with 
radius  r,,  as  £>/,  referred  to  the  origin  O  are:  OH/  =  lt  and  #/£>/  =  &,. 
and  of  Z>//  O'H/f  =  I,,,  H,,D,t  =  k,,;  whence 


=  tan  (a,  +  J8.)  ;  (11) 


I,,  —  xt,  —  rlt  sin 
/?//  =  rt,  +  w//, 
whence 


< 


or  if  we  have  the  value  of  / 


THE    SPIRAL  7 

from  either  of  these  equations 

0,,°  =  a,,0  -  (a,°  +  8,0  ),  (14) 

with  £//°  reduced  to  arc,  then 

r/A/  -  AnBn 
and 

£/  =  «//  -  (<*/  +  /3,,),  r,S,  =  A,B,\  (15) 

denoting  the  difference  between  L  and  AnBfl  +  A,B,  by  "d"  we 
have 

d  =  J/  -  U//5//  +  ^4/B/)  =  L  -  (r^jS,,  +  r^) 

Placing 

(r//^//  +  r,/3,)  =  Lct 

we  have 

"d"  =  L-Lc.  (16) 


§5.   TRANSFORMATION   TO   RECTANGULAR   CO-ORDINATES. 

By  the  Calculus  dy  =  dL  sin  ^ 

rfx  ==  dL  cos  ^  ; 

. 

in  which  <j>  =  any  angle.     By  trigonometry, 


*  This  and  the  succeeding  formula  are  for  expressing  the  trigo- 
nometrical function  of  an  arc  in  terms  of  the  arc  itself  (see 
Chauvenet's  Trigonometry,  Chapter  XIII,  1867). 

m  and  n  —  positive  whole  numbers,     m  =  even,  n  —  odd. 


8  TRANSITION    CURVE 

in  which  m  may  have  any  value  from  zero  to  infinity,  n  may  have 
any  value  from  one  to  infinity,  and  <£  any  value  from  zero  to  infinity. 
Substituting  in  the  equations  for  sin  0  and  cos  <£, 


we  have 

-  =  sin  ^  =  sin  (£,/  +  £,)  =  sin  B/T 


(I7W 


EXAMPLE  No.  2.     Given   L  =  150;    r,,  =  818.8;   r,  =  2865;   to 
find  <£. 

sin  ^  =  sin  O  +  ^)  -          i-  +  -- 


L         24  Vr^       r//  J 

150/1  1    \fi       (150)  V    1  1    \ 

sin  0  -  sin  &„  +  0,)  -  3^,4-  28^)  [1  -  -24-(8l878  +  2865) 

sin  ^  =  sin  O//  +  &,)  =  .11775  X  .9977  =  .11748. 


If  we  make  r/  =  oc,  —  =  0; 


we  have,  with  origin  at  O,  $  =  a,, ;  and  L  =  L//  (17ft)  becomes 


l*(l  -  ±*L  +  _^1_  _  etc.)  _  sin  Bl,DllNll     (I7C) 


THE    SPIRAL 


_     __  _ 

2P        48P3         3840P5 


i»  \ 

n  )  : 
A'PV 


-etc.  ...  —  );         (18) 


integrating, 


L 
or,  since  P=  J 1,  with  its  origin  at  B-t  the  ordinate  of  any 

TI,  Ti 

point  of  the  spiral,  as 


EXAMPLE  3.     Given  r,  =  2865;    r/,  =  818.8;   L  =  150,  to   find 
V  by: 


i 


-.  -.00035)  [1  -  (.00122  -  .00035)2  +  etc.] 

D  OO 


y  =  22500  X  .00087 
=  3.26225. 


Or  in  terms  of  <b,  since  L  =»  2r^f  and  —  =  ( ) 

*"       \  r,/      r// 


-fj-Hetc.  I.     (20a) 


10  TRANSITION   CURVE 

If  we  make  r,  =  oo,  —  =  0,  L  =  L,,  the  ordinate  of  any  point 
of  the  spiral,  as 

B//G//  =  y//  =  |p  (l  — ^7  +  etc.)     with  origin  at  O       (21) 

and  OG//  as  abscissa. 
Similarly, 

etc.  (22a) 


integrating, 


x  -L  [  1-  4Qp2  +  3456p4  -  599Q40P6  +  6tC 


(m  +  1)  MP* 


(23) 


With  origin  at  B/  then  the  abscissa  of  any  point  of  the  spiral,  as 

KB,,.-*-  L  [l-  ^-  (-  -  ^-)2+^  (i-  i)'  -  etc.     (24) 
L        40   \r/,        r//       3456  \r//       r// 

or  in  terms  of  <j>,  since  L  =  2r<£,  (25) 


x  is  laid  off  on  the  arc  of  the  circle  A/B/  or  AnBn  (with  radii  r,  or 
r//)  as  axis  of  X  with  J5/  or  Bt,  as  origin,  x  being  the  abscissa  of  y 
which  is  laid  off  normal,  or  radial,  to  the  arc  A,B,  or  A  //#//. 


THE   SPIRAL  11 

EXAMPLE  4. 

x  =  L  [l  -  ^  (^  -  ^)2±  etc.  (other  terms)  ] 

/         22500  X  .0000007569  +   .  \ 
U  \  40  /  ' 

x  =  150  (.9996  ±)  =  149.94. 
With  origin  at  O,  any  abscissa,  as 


also  when  <j>  =  <*// 

^=2^(1-^  +  ^  -etc.)  (26a) 


§  6.   RECTANGULAR   CO-ORDINATES   AT   THE   OFFSET 
DISTANCE. 

i°.   To  find  the  abscissa  and  length  of  the  curve. 

Let  j/»»//  be  the  ordinate,  xmt/  the  abscissa,  and  L*»/,  the  length 
of  the  curve  corresponding  to  xm^ym/r,  then  a  very  nearly  correct 
value  of  ym-n  for  central  angles  up  to  50  degrees  may  be  obtained 
as  follows: 

OH,,  =  xm,i  =  xu  —  rlt  sin  ^>:  whence,  with  origin  at  O,  (27) 

by  Eq.  *  page  7  and  (27). 

OH,,  =  xmtl  =  r//4>(l  —  —  +  — ...  .J  or  (27a) 

in  terms  of  r,/  and  L// 

OH,,  =  x*«  -  —  (l  -      L//22  +       L//4       )       (28) 

and  when  the  origin  is  at  Bi 

in  which  L  =  I/,,  —  L,  =  BtBu. 


12 


TRANSITION    CURVE 


A  close  approximation  to  Lf  or  Lm  is  La  which  is  determined  by 
making 


La- 


••  L  +  Wxf 
12 


whence 

with  origin  at  B/,  then 

*  BtU  =  Lf  = 


40  Va       rj 


(29) 


.      (30) 


We  may  write  Lm  for  Lf  and   xm   for  xf,  rm  for  ra,  —   =  0.  in 

TI 
equation  (30)  when  the  origin  is  at  O;  then 


1  - 


La-' 


(30a) 


2°.   To  find  the  ordinate  of  a  transition  curve  measured  on  the 
offset  m//  (or  f): 


Fig.  2a. 


*  After  solving  (Eq.  30)  for  Lf,  compute  rf,  give  La  and  ra,  in 
the  denominator,  the  value  just  found  for  Z//and  rf,  and  solve  again 
for  L/till  the  Eq.x/=L/  Fl  -  ^~  (j^  -  ~}~  etc.,  is  true,  ^/having 
a  fixed  value  from  Eqs.  (27-28a). 


THE    SPIRAL  13 

The  method  given  in  Equation  (47)  for  finding  the  ordinate  y  / 
is  approximate  and  sufficiently  close  within  the  limits  of  the 
tables.  For  a  large  central  angle  or  great  length  of  spiral  a  closer 
approximation  is  necessary. 

Writing  ym,,,  for  y,  when  origin  is  at  O,  we  have  by  (19)  : 


-  )'<»> 


with  origin  at  B,  then  B,U  =  Lf  =  P  (-  --  —  )  ,  and 


This  method  of  obtaining  £//  and  j8/  may  be  used  instead  of 
Eq.  6  or  8. 


§  7.    OFFSET  DISTANCE. 
In  general,  by  Fig.  (2),  the  offset  distance. 

/  =  y  -  R  ver  <j>  =  y  -  R  (1  -  cos  <j>\  (33) 

in  which  R  depends  on  —  —  —  for  its  value,  substituting  for  cos  $ 
its  value  (remembering  that  R  =  yj»  and  reducing,  since 

R  (1  -  cos  #  =  R  [l  -  (l  -  j~    +  3^  -  etc.)]' 
/  =  y  -flver^  =  L  [(^  -  ^  +etc.)- 

(i- 31^3+ etc.)];  (34) 


14  TRANSITION   CURVE 


-:    (35) 


since  p  =  f  —   --  V  we  have  for  (36) 

the  offset  with  origin  at  B,\ 


EXAMPLE  5.  Given  r,,  =  818.8;  r,  =  2865;  L  =  150;  to  find 
the  offset  /.  First,  when  the  circular  curves  r//  and  r/  turn  in  the 
same  direction,  by  (125): 


- 
24         "  r      L         112  W  ~  i 


X  .00087  [l  - 


(150)2  (.00087)2 


24  112 

With  origin  at  O,  when  <f>  becomes  «//,  L  becomes  L//  and  f  becomes 
m//,  then 

*  N"H"  -  m-  -  T£,  (l  ~  i£r?  +  etc- )         (38) 

Second,  when  the  circular  curves  turn  in  opposite  directions  or  are 
reversed. 


If  L,,  =  L/,  then 


For  the  distance  between  the  centtrs  of  the  two  curves,  turning 
in  the  same  direction,  we  have: 

r/-  (r,,HKf)-W>,/-r,-J*w  +^(^  -  ^)d  -etc.)].     (41) 


*  For  the  constant  P,  when  OB,   =  B,B,,  or  L,  =  L,  then  H,U 
-4/f/,  m/  =  /,  and  5/G/  =-  J5//5///,  or  y,  =  2/  =  5/£4  (nearly). 
t  In  this  case  there  will  be  two  values  of  P  except  when  r/,  =  r/. 


THE    SPIRAL  15 

EXAMPLE  6.  Given  L  =  150;  r/  =  2865;  r/,  =  818.8;  to  find 
the  distance  between  the  centers  of  curves  with  radii  r//  and  r,, 
First,  if  curves  turn  in  the  same  direction  (Fig.  2): 


£>„£>,  =  r,  -  (r/,  +  /)  =  2865  -    siS.S  +     —     +  .00087    ; 

D«D,  =  2865  -  [818.8  +  .815]  =  2865  -  819.615  =  2045.385. 

Second,  if  curves  are  reversed  (or  turn  in  a  contrary  direction) 
(Fig.  9),  then: 


X  sec  K,,D,D,,t  =  [r/  +  (r//  -f  mtl  +  m/)]  sec  KltDitD,,,; 

tan  K,tD,Dtl,  =  »»"+**'*  --  (416) 

r/  +  TH  +  m//  +  m/ 


PRACTICAL    FORMULAE. 

§  8.   If,  in  the  foregoing  values  of  y,  x,  -pf,  and  /  we 

dLi 

omit  terms  in  the  bracket  after  the  first  or  second,  we 
have  for  central  angles  of  20°,  or  less,  the  following 
practical  formulae  for  uniting  circular  curves  of  differ- 
ent radii  by  means  of  a  spiral  arc  : 


See  Eq.  (32) 


See  Eq.  (28a) 


*  When  0,O,,  =  0. 


16  TRANSITION    CURVE 

B.B,,  =  L  =  x  ( i  +  ~p,J  ;  nearly  (44) 


sinB/WB,,  =  sin0  =sin  (pit+  ft)  =  (44!)) 

L/  i        i\f          L2/  i        i  V 

-[  —  +-      i  --  —  +   -  }  +  etc. 

2\T,,      rJL          24Vr/7       rj 

arcByWB/x  (atradius  =  i)  =  -(  ^-+  ^-)  =  arc  0:     (440) 

2  \r//     r/  / 

arc  0  X  57-3°  -  0° 

A'A"=f=gfe-y,  (45) 

from  which  latter  equation  we  find: 


identical  with  Froude's  curve  of  adjustment,  as  indicated 
by  Professor  Rankine  ("  Civil  Engineering,"  Edition 
1863). 

|  /=  —  (-  ---  ),  and  if  in  the  value  for  y  we  write  JL 
48  \TM      Ttj 

for  L  and  cube  it,  then 

WL_u_u_  .  =  JL=vsL_i\ 

6P         8_     48P      4SR       48\r/;      rj'    v     ; 
6P 

Hence  the  ordinate  y  =  J  f  is  at  the  middle  point  of 
the  transition  curve. 


THE    SPIRAL  17 

For  uniting  a  tangent  with  a  circular  curve  by  the 
use  of  the  spiral, 

we  have  —  =  o,  in  (42-47); 

whence:  by  Fig.  2,  for  any  distance  on  the  spiral,  as  L,,, 
with  radius  rlly  the  corresponding 

Ordinate  Gy/By/  =  y//  =  ^  (48) 

r        L  2     i 

and  the  abscissa  OG,,  =  xy/  =  Ly/     i  -  -—~-2+       (49) 

L  4or//        J 

The  sine   of  central  angle  of  the   transition  curve 
equals 

L      /    "       L  2  \ 

sinN^D^B^  =  sin  a//  =  -^  (  1  -  ^-^-J       (50) 
2r//  \          24r/y  / 

arc  N^D^B^  =  a//  =  -^- ,  arc  a/7  X  57-3°  =  «//° 


the  offset  N/yH/x  =  my/  =       -  (51) 

241  tl 

\  the  length  of  spiral  =  —  ^  =  Vbm^r^         (52) 


OH,,  =  xm//  =  x/x  -  ry/  sin  ay/,  or  (a) 

(sab) 


Oyd  or  0,1  =  t=  xm+  (ry/  +  m)  tan  \  I  (see  Fig.  5)  (6) 

If  D//  and  Z>/  denote  the  "  degrees  "  of  the  curves  (determined 
by  100  feet  of  their  length)  corresponding  to  the  radii,  r//  and  r/; 
then,  since  L//  and  Z>//  and  Lf  and  Z>/  are  inverse  functions  of  r// 
and  r/,  we  have 

r         r        r        T     D"  ~  D'       A    1        l       L"  (D"  ~  D'\ 
L,,-L,=  L  =  Ltt        D//       and  --  -=  —  (—j^-) 


18  TRANSITION    CURVE 

TO  LAY  OUT  THE  TRANSITION  CURVE  BY  DEFLECTIONS. 

§  9.  If  at  the  point  Bt  (see  Fig.  2)  we  imagine  r,  to 
increase  until  it  becomes  infinitely  great,  the  curvature 
of  BlAi  =  0  and  the  arc  BtA.t  will  be  a  straight  line  still 
preserving  its  tangency  to  the  transition  curve.  The 
curvature  at  Btl  will  diminish  to  the  same  extent,  i.e., 
the  difference  between  curvature  at  B/  and  Btl  will  be  the 

L2 
same  as  when  $,  =  0,  and   $u  =  op  •    The  ordinates  x 

y  can  be  computed  and  laid  off  from  the  new  tangent  as 
axis  of  abscissa  with  Bt  as  origin,  the  same  as  if  from  0  *. 
If  we  now  conceive  this  new  axis  of  x  to  be  curved 
to  a  radius  r,  the  curvature  of  the  transition  curve  at  any 
point  will  be  increased  by  the  same  amount  and  the 
ordinates  may,  without  serious  error,  be  laid  off  normal  to 
the  arc  B,A,  and  establish  points  of  the  transition  curve. 
The  same  reasoning  will  apply  if  >/y  =00  and  BtlAlt 
becomes  tangent  and  values  of  x  and  y  be  laid  off  from 
it  with  Btl  as  origin,  except  that  the  resulting  transition 
curve  would  be  convex  to  BtlAlt.  The  ordinates  would, 
however,  be  equal  to  those  of  the  corresponding  distance 
from  Br  If  rn  now  resume  its  original  length  the  cur- 
vature of  the  transition  curve  at  any  point  will  equal 
that  of  the  circular  curve  with  radius  rf/  minus  the  cur- 
vature it  had  in  a  contrary  direction  when  ru  was 
infinitely  great  and  BtlA.lt  a  straight  line.  In  determin- 

p 
ing  /3y/  and  r  =  -=- ,  data  may  be  taken  from  the  tables. 

Li 

Equations  for  x  and  y  are  equally  true  whether  the 
origin  be  taken  as  0,  Bn  or  Blt. 

x  will  be  measured  on  A,B/t  and  y  normal  to  AtBf. 
To  lay  the  transition  curve  out  by  the  expressions 

*  In  Fig.  2,  O,  should  be  marked  B.  S.  or  E.  S.  and  B,  or  Bi, 
marked  B.  C.  C.  or  E.  C.  C.  in  staking  out  a  curve  on  the  ground, 
according  to  the  direction  in  which  the  line  runs. 


THE    SPIRAL 


19 


for  x  and  y,  their  values  may  be  laid  out  simultaneously 
with  corresponding  equal  chord  measurements  along  the 
transition  curve. 

The  principle  enunciated  in  the  paragraph  preceding,  enables 
us  to  prepare  a  table  of  deflection  angles  according  to  the  following 
method : 


<K 


Referring  to  Fig.  3.  If  the  deflection  angles  from  AX  to  any 
point  be  denoted  by  SS,S/,,  it  will  be  found  by  computation  that 
any  angle  as 

DAX  =  5,  =  -  =  gp-  (nearly),  *  (53) 

in  which  r,,  =  DK  and  L  —  AD. 

ADC,  =  DCX  -  DAX  =  a,-^=^--g^-=—  =.     (54) 


DD,        y,,  a, 

*  Since  -jyr-  =  —  =  tan  —  (nearly). 

AL/i  Xff  O 


20  TRANSITION    CURVE 

the  angle  which  CT,  a  tangent  common  to  the  spiral  and  circular 
curve  with  radius  r//,  makes  with  the  chord  AD  produced.  To 
establish  the  points  E,  F  and  G  by  deflection  at  D,  from  tangent 
DT  we  have,  from  the  paragraph  already  referred  to, 

EDT  =  5  +  A,  (55) 

and  8  =  the  deflection  from  AX  to  B,  and  A,  =  the  deflection 
from  DT  to  the  point  E,  for  the  circular  curve  with  radius  r,,.  In 
the  same  way  with  D  as  origin. 

GDT  =  8,,  +  A//;  =  a.  (56) 

The  angle 

DGG,,  =  2  «,/  +  A,,  =  (a  „-  a,)  -  (S,,  +  A,,),  and         (57) 
if  we  add  (5,,  -|-  A,,)  to  both  members  of  the  equation,  we  have: 

GOT  =  a,,  -  a,  =  ^  =  3  fi,,  +  2  A//t  (58) 

in  which  A,,  =  LA,  A  =  the  deflection  for  a  unit  length  of  Z>(r  on 
the  circular  curve  with  radius  r//,  and  L  the  units  from  D  of  any 
point  laid  off  on  DG. 

FOR  CONVENIENCE  IN  FIELD  WORK. 

Eq.  (53)  may  be  reduced  to  degrees  and  put  in  the 
following  form : 

5°  =  —  57.  3°,  whence  by  Eq.  (£6) 
or 

*  .     o>°  =  DAL  ±  ^-  57.3°  in  which  (59) 

or 

the  instrument  point  is  the  origin  of  L. 

A  =  deflection  angle  per  foot  from  tangent  for  1° 
circular  curve. 

D  ==  degree,  or  rate  of  curvature,  at  position  of  in- 
strument. 

D,  =  the  degree,  or  rate  of  curvature,  at  the  point  to  be 
located. 

w  =  deflection  from  tangent  at  any  point  of  spiral  to 
locate  any  other  point  of  spiral  (using  the  +  sign  for 
running  toward  G,  —  sign  towards  A. ) 


*  More  nearly  57.2956° 


THE    SPIKAL  21 

This  equation  can  be  still  further  simplified  in  appli- 
cation by  the  following  reduction  in  5°  =  —  57.3°,  let  r 

=  radius  due  to  L>  measured  from  the   position  of  the 
instrument  to  the  point  of  spiral  to  be  located. 

N  =  the  number  of  chords  of  equal  length  (each  sub- 
tending one  degree  change  in  rate  of  curvature)  in  the 
length  of  L  of  the  spiral.  Then  if  r0  =  the  radius  of  a 
one-degree  curve, 


r  =  ^  and  5°  =  ^J^T    =  LN  *  0.00166°     (60) 
1\  O/oU 

or  8f  =  LN  0°00.1'. 

If  we  wish  the  change  in  rate  of  curvature  to  be  any 
other  than  one  degree  to  the  chord,  and  denote  this  change 
by  C,  C  =  the  change  in  degree  of  curvature,  due  to 
one-chord  length,  and  may  be  either  a  fraction  or  whole 
number;  then 

8'  =  LNC  X  0°00.1';  (60a) 

or  since  C  =  ^  ,  8'  =  LD,  x  0°00.1'  ;*         (606) 

and  if  one  of  the  chords  be  fractional,  and  we  indicate  the 
fractional  part  by  -=  in  which  F  is  the  number  of  parts 

into  which  a  chord  is  divided,  and  n  the  number  of  such 
parts  taken, 

8'  =  L(N  +  j\  C  X  0°00.1',  hence  (60c) 

r          /         n\  1 

w'  *  L  I  DA  ±Ylf  -h  =|C  X  o°oo.i'  (61) 

is  a  general  formula  for  the  deflection  from  a  tangent, 
at  any  point  of  the  transition  curve,  to  locate  any 
other  point  of  the  transition  curve. 

*  See  Appendix. 


22  TRANSITION   CURVE 

J)     T) 

In  which  case  C  =  — » 

to  be  computed  and  substituted  in  (61). 

If  n  =  o,  C  =      '" 

and  «'  =  L  [DA+  (D,  -  D)  o°oo.i'] 

With  instrument  at  A,  Z)A  =  0. 

To  place  the  line  of  sight  on  tangent  at  any  point  of  the  spiral. 
After  backsight  on  last  instrument  point,  formula  (61),  for  deflec- 
tion to  tangent  becomes 

a>'  =  L  [z>A+  2  (N  +  j^  C0°00.1'l 

when  running  towards  G  (Fig.  3),  D  being  the  rate  of  curvature  at 
the  last  instrument  point,  and  L  the  length  of  spiral  to  backsight. 

See  Eq.  (57)  of  =  L  \D±  -  2  \N  +  j\  C 0°00.1'J  running  towards 
A. 

In  applying  Eq.  (61),  the  more  frequent  the  change  points,  the 

more  nearly  will    the  resulting   curve  agree   with  the  theoretical 

spiral  ;  their   distance  apart   to  be    not  more  than  150  to  300  ft. 

nor  '*  change  points  "  include  a  central  angle  of  more  than  10°  to  15°. 

A  nearer  value  for  the  second  term  in  the  bracket  of  (61)  is: 

(N  +  |;)  C  0°  00'.0998. 

EXAMPLE  7.     Given,   D  =  2,  A  =  .3',  L  =  125,  N  =  2,  ~  =  J, 

r 

C  =  1,  Chords  =  50  ft.,  to  find  the  deflection  from  a  tangent  at  C 
to  locate  a  point  E  +  25  (see  Fig.  4).     By  formulae  (61): 

at'  =  L  [jDA  +   (.V  +  fy  X  1  X  0°00.1'1 

substituting  values  given  above, 

<u'  =  125  [2  X  .3'  +  (2  +  i)  X  1  X  0°  00.1'],  or 
a'  =  125  [.6'  +  .25']  =  125  X  .85'  =  1°  46i'. 

(See  table,  Spiral  1.) 

The  foregoing  values  for  the  deflection  angles  are  closely  approxi- 
mate, though  the  method  indicated  in  connection  with  Fig.  4  in 
determining  any  deflection  angle,  as  fin,  while  less  elegant  is  more 
nearly  exact. 

By  Eqs.  (21)  and  (26)  tan  fin  =  ^,  (62) 

from  which  we  determine  fin. 


THE    SPIRAL 


23 


Fig.  4. 

"  n  "  having  any  value,  whole  number  or  fractional. 

With  A  as  origin  we  obtain,  by  (62)  the  deflection  angles  to  the 
points  B,  C,  D,  E,  etc.,  to  any  change  point,  as  C,  where  the  degree, 
or  rate  of  curvature,  is  D.  From  C,  with  backsight  on  A,  deflect 
a.  —  Sn  (in  which  n  =  2,  and  a  ==  the  central  angle  of  the  spiral  from 
A  to  C)  to  get  on  tangent  at  C,  then  to  locate  any  point,  as  E  (not 
shown  on  Fig.  4),  from  a  tangent  at  C,  deflect 

5n  (63) 


£n  being  the  deflection  angle  due  to  n  chords  from  C,  corresponding 
to  the  same  number  of  chords  from  A. 

To  get  on  tangent  at  E: 

With  backsight  on  C,  deflect  I/Z)A  +  (a  -  &»),  in  which  n  =  2, 
L  =  CE,  and  a  corresponds  to  the  central  angle  of  the  spiral  from  A 
for  a  length  L. 

The  process  is  the  same  as  with  Eq.  (61),  though  the  second  factor 
of  the  second  member  is  different. 


*  It  will  be  seen  that  the  differences  between  this  method  and 
that  of  Eq.  (61),  for  a  curve  with  L  =  400  ft.  and  8  =  16°  is  0°00'03". 

The  difference  increases  with  the  central  angle  a.  The  above 
method,  with  change  points  200  to  300  ft.  apart,  is  quite  accurate 
arid  the  best  for  preparing  a  set  of  tables  —  though  not  so  easily  ap- 
plied in  field  computations  as  (60)  or  (61).  By  assuming  "change 
points  "  100  to  250  ft.  apart,  the  deflection  tables  of  this  book  may  be 
extended  indefinitely. 


24  TRANSITION   CURVE 


The  several  angles  can  be  computed  and  tabulated,  to  any 
number  which  is  likely  to  be  needed,  to  conform  to  any  system  of 
"change  points"  determined  upon  after  #o1/o,  etc.,  have  been  com- 
puted for  the  particular  transition  curve  where  value  of  P  has  been 
fixed  in  conformity  with  the  character  of  the  alignment. 

EXAMPLE  8.  Given  L  =  200;  xc  =  199.91;  yc  =  4.65;  to  locate 
C  from  A. 

tan  Sc  =       6       =  .023260  or  fifl  =  1°  19.95'. 


To  get  on  tangent  at  C,  at  which  point  the  total  curvature  is  4° 00': 
4°  00'  -  1°  19.95'  =  2°  40.05'.  Hence  with  the  instrument  at  C  and 
backsight  on  A,  deflect  2°  40.05'  to  get  on  tangent  at  C,  where  the 
rate  of  curvature  is  D  —  4°  00'. 

From  tangent  at  Cto  locate  some  point  E,  which  is,  in  this  case, 
200  ft.  from  C,  then 

01  =  LDA  +  8n  =  200  X  4  X  .3'  +  1°  19.95'  =  5°  19.95' 
and  to  get  on  tangent  at  E,  with  backsight  on  C, 

o)  =  LDA  +  (a  -  5C)  =  200  X  4  X  .3'  -f  2°  40.05'  =  6°  40.05'. 

ORDINATES. 

§  10.   To  determine  the  ordinates  o,   01,   etc.     From  any  chord 
as  Z0,  let  aB  =  o,  Bib  —  s,  AB^  =  XQ,  BiB  =  y0,  bABi  =  y. 
Then  from  Fig.  4,—  =  tan  y,  s  =  XQ  tan  y,  s  —  y0  =  XQ  tan  y  —  yQt 

XQ 

—  =  cos  y,  o  =  (s  —  yo)  cos  y,  or  since  s  =  XQ  tan  y, 

o  =  (x0  tan  y  -  y0  cos  y  =  x0  sin  y  -  y0  cos  y  :  (64) 

For  the  distances  Aa,  etc.,  Aa  =  A±Bi,  sec  y  —  aB  tan  y  =  XQ 
sec  y  —  o  tan  y.  In  the  same  way  we  may  obtain  ot  and  Aa\.  To 
determine  the  point  B,  C,  etc.,  by  measurement  alone:  First  com- 
pute and  lay  off  the  distances  Aa,  Aai,  etc.,  then  lay  off  AB  and  aB 
simultaneously;  next  BC  and  a\C,  etc.;  when  y  is  small  the  dis- 
tances bB  and  biC  may  be  used,  distances  A b  and  Abi  being  com- 
puted and  laid  off  first. 

As  a  check  it  will  be  an  advantage  to  compute  the  length  of  the 
long  chord  as  well  as  the  angle  it  makes  with  the  axis  of  x,  thus: 


;  (65) 

in  which  (n)  equals  the  number  of  increments  or  stations  between 


THE    SPIRAL  25 

A  and  D.     Let  the  length  of  any  chord  be  c2,  c4 ,  en,  in  which  2,  4,  n 
indicate  the  2d,  4th  and  nth  increment. 

c2  =  (x2  —  EI)  sec  y2;   c4  =  (.r4  —  z3)  sec  Y4; 

c»=  (*n  -  xn-i>  sec  yn-  C66) 

EXAMPLE  9.  Given  x2  =  209.65;  xv  =»  120;  y2  =  4°  37',  to  find 
the  length  of  chord,  c2=  (x2-  a*)  sec  72  =  89.65  X  1.0032  =  89.93. 
For  the  length  of  long  chord  (from  origin),  x2  =  209.65;  7=2°  27'; 
x2  sec  y  =  209.65  X  1.009  =  209.84. 

It  is  to  be  observed  that  the  superelevation  of  the 
outer  rail,  in  the  use  of  the  transition  curve,  may  be 
made  greater  or  less  than  that  which  has  been  assumed 
in  computing  the  tables;  the  only  effect  it  will  have  is  to 
diminish  or  increase  the  assumed  value  of  "i"  which  is 
equivalent  to  increasing  or  diminishing  the  velocity, 
since  i  and  v  are  inverse  functions  of  each  other  in  the 
constant  P,  i.e.,  it  makes  the  rate  of  rise  of  the  outer 
rail  to  effect  superelevation  a  little  greater  or  less.  It 
is,  however,  best  to  introduce  the  average  velocity  of  the 
express  or  fast  passenger  trains  in  constructing  the  tables. 
Where  the  location  is  so  constrained  that  the  EC's  and 
BC's  of  the  circular  curves  are  quite  close  together,  it 
may  be  necessary  to  give  "i"  a  smaller  value  than  would 
be  otherwise  desirable.  A  value  of  300  or  400  is  suffi- 
cient for  adjustment,  and  good  results  may  be  obtained 
with  a  value  of  200  when  the  radius  is  not  greater  than 
573  feet,  since  v  usually  is  made  to  decrease  with  r. 

The  beginning  and  end  of  the  transition  curve  should 
be  marked  by  permanent  points. 


26 


TRANSITION   CURVE 


Degree 
of 
Curve 

Radius 

Reciprocal 

Degree 
of 
Curve 

Radius 

Reciprocal 

D 

r 

1 
r 

D 

r 

1 
r 

0°30' 

11460. 

.00008726 

8°00 

716.3 

.00139626 

1°00' 

5730. 

.00017453 

9°00 

636.7 

.00157079 

1°30' 

3820. 

.00026179 

10°00 

573. 

.00174533 

2°00/ 

2865. 

.00034906 

11°00 

520.9 

.00191986 

2°30/ 

2292. 

.00043633 

12°00 

477.5 

.00209439 

3°00' 

1910. 

.00052359 

13°00 

440.8 

.00226893 

3°30' 

1637.1 

.00061086 

14°00 

409.3 

.00244346 

4°00' 

1432.5 

.00069808 

15°00 

382. 

.00261799 

4°30' 

1273.3 

.00078540 

16°00 

358.1 

.00279253 

5°00' 

1146. 

.00087267 

17°00 

337. 

.00296706 

5°30' 

1041.8 

.00095993 

18°00 

318.3 

.00314159 

6°00' 

955. 

.00104712 

19°00 

301.6 

.00331613 

7°00' 

818.8 

.00122173 

20°00 

286.5 

.00349066 

THE   SPIBAL 


27 


PROBLEM  I. 
TO  FIND  THE  SEMI-TANGENTS. 

§  ii.  Given  a  circular  curve  whose  radius  =  rtl\  the 
intersection  angle  =  /;  the  semi-tangent  =  T,  to  unite 
it  with  the  tangents  by  means  of  transition  curves  whose 
lengths  are  Lt  and  L  and  offsets  are  mt  and  m  respectively. 


flr.  5. 


CASE  1. 
When  my  >  m  (by  Fig.  5),  if  I  <  90°, 


«=  ^m/  -}-  T7  —  w/  cot  7  +  m  cose  7. 


28  TRANSITION   CURVE 

If  7  >  90°, 

O,d  =  0IHI  +  N,c  +  ab  +  bd  = 

xm/  +  T  -  m,  (-  cot  7)  +  TO  cose  7;  (2) 

0,d  =  xm+  T  +  m/  cot  7  +  m  cose  7;  (3) 

or  in  general  calling, 

O^d,  or  O,d  =  t,  ;  t,  =  xm/  +T  =p  m,  cot  I+m  cose  I,  (4) 

the  +  sign  being  used  when  7  >  90°  and  the  —  sign  when 
7  <  90°. 

If  7  <  90°,  0,d,  =  OIHI  +  AT,c,  +  c,&,  -  d,gr,  = 

#m  -f  T7  +  m,  cose  7  —  m  cot  7;  (5) 

If  7  >  90°,  0,d  =  (),#,  +  N,c  +  cb  +  dg  = 

^m  +  T7  +  my  cose  7  +  m  cot  7.  (6) 

In  general  calling, 

O.d,  or  O,d  =  t;  t  =  Xm  -f  T  +  m,  cose  I  ±  m  cot  I,     (7) 
using  +  when  7  >  90°  and  -  when  7  <  90°. 

CASE  2. 
If  m  =  o  in  equation  4,  we  have 

t,  =  T  +  m,  cose  I,  t  =  XM  +  T  +  m/  cot  I.  (8) 

CASE  3. 
If  m  =  m  in  equation  4, 

tt  =  zm/  +  T7  4-  TO,  (cose  7  ±  cot  7),  = 

£    =  zw  +  !T  +  m  (cose  7  ±  cot  7).  (9) 

If  7  <  90°,  the  last  term  of  t,  is  m,  (cose  7  -  cot  7)  and 

by  trigonometry  (see  Chauvenet's,  p.  35). 

,  T         I          cos  7      1  —  cos  7  ,  _ 

cose  7  —  cot  7  =  -. — r : — -r  =  — : — = —  ==  tan  J  7 

sin  1       sin  jf  sm  L 

.'.  tt  =  xm/  +  T  -f  m,  tan  i  7 

or  ^  =  xm/  +  (r,,  +  mt)  tan  }  7  ;  (10) 

similarly,  if  7  >  90°, 

T  ,    T         1    +   COS  7 

cose  7  +  cot  7  = : — = —  ; 

sin  / 


THE    SPIRAL  29 

but  if  /  >  90°, 

cos  /  =  —  cos  /  and  cot  /  =  —  cot  7 

T  ,  .    TN  1    —   COS  7 

cose  7  +  (—  cot  7)  =  -  :  —  7  —  , 
sin  7 

which  we  have  seen  =  tan  J7.     Hence  when  my  =  m, 

ty  =t  =xm  +  T  +  m  tan  JI  =  xm+  (r,,  +m)  tan  JI     (n)* 
(See  App.  I) 

is  a  general  equation  whether  7  be  greater  or  less  than 
90°.  If  in  (11)  we  make  xm  and  m  each  =  o,  then 
£  =  T  =  r/x  tan  J7  =  the  semi-tangent  for  a  simple  circu- 
lar curve. 

EXAMPLE  10.     Given  L  =  210;  r  =  818.8;  m/  =  2.23;  /  =  40°; 
O/#/  =  104.93;  to  find  t  (Fig.  5): 
*  =  *m/  +  (*•"  +  w/)  tan  £7  =  104.93  +  821.03  X  .36397  =403.75 


With  the  same  data 

§  12.    TO  FIND  THE  EXTERNAL  SECANT. 

CASE  1. 
When  m,  >  m, 

H.d,       T  +  m,  cose  7  +  m  cot  7 

^ry    =  -  =  tan  ri.Dd.  ; 

DH,  ru  +  m 

eA  =  (ry/  +  m)secH/Dd/  -  ry/.  (12) 

*  If  we  wish  to  unite  two  grades  by  a  vertical  curve,  then,  if 

in  Eq.  44,  we  make  —  =  o 

TI 
and  sin  <£  =  sin  £7 

L  =  2r//  sin  £7 
*  -  L  (1  -) 

1,2 

y=    6^, 
and  Eq.  11,  Prob.  1  t  =  xm  +  (r//  +  m)  tan  £7 

in  which  £7  =  J  the  angle  at  which  the  grades  intersect,  r//  should 
have  a  value  of  from  5730  to  11460,  and  represents  the  minimum 
radius  of  curvature  at  the  middle  point  of  the  vertical  curve,  x  and  y 
ordinates  from  the  origin  or  terminus  of  the  vertical  curve  to  its 
middle  point  or  to  any  other  point. 


30  TRANSITION   CURVE 

CASE  2. 
If  m  =  O, 

N,b,      T  +  m,  cose  /  ,  r  ~, 

~  =  -      — ^r =  tan  A^  ; 

e^b,  =  r,,  sec  N,Db,  -  ry/.  (13) 

CASE  3. 
m,  =  m,  e.d,  =  (r,,  +  m)  sec  JI  -  r,,  (14) 

If,  in  Eq.  14,  we  make  m  =  o,  we  have  for  the  external 
secant  of  a  simple  circular  curve : 

ed  =  rit  (sec  \l  —  1)  =  ec. 

If  we  wish  the  transition  curves  BUiOul  and  Bf)u  to 
become  a  tangent  to  each  other  at  some  point  A  with  a 
minimum  radius  of  curvature  =  ru  =  the  radius  of  the 
circular  portion  BltlBn  with  B,4,B,  elided >  then  BUI  is 
coincident  with  B4. 

B,DN,  -«  =  /  -  A-^ZXY,,  =  a/  =  L-— ,  (15) 

a,  +  a  =  /  (16) 

0///B///=L/=2r//(I-  «) 


O/B/  =  L  =  2r/y  (I  —  a,)  =  v  24T,,m  (18) 

NlfHtt-mt  =^  (19) 

L2 


AV 


94., 
rifi*t  //y 


If  77^/  =  m  then  L/  =  L  =  2r/y  J7,  the  point  A  is  at 
the  middle  point  of  the  curve  and  L,  —  J  the  total  length 
of  the  curve  from  £.£.,  through  C.  C.  to  #.  S. 

Since  ax  and  a  are  expressed  in  arc  in  the  above  for- 
mulas, while  the  intersection  angle  is  measured  in  degrees 
and  minutes,  the  latter  will  have  to  be  reduced  to  arc  at 


THE    SPIRAL 


31 


radius  =  1  before  entering  the  formulas,  and  after  calcu- 
lations are  made  the  result  expressed  again  in  degrees 
and  minutes  to  apply  formulas  for  semi-tangents,  ex- 
ternal secants,  etc. 


PROBLEM   II. 
TO  FIND  THE  LOCATION  OF  THE  OFFSET  "/". 

§  13.  Given  two  curves  with  radii  r,  and  rin  a  distance 
D,D,,  =  d  joining  the  points  D,  and  Dlt  also  the  angles 
BD,D,,  =  p  and  CD,,D,  =  0,  to  find  the  points  A,  and 
A(l  at  which  a  line  drawn  through  the  centers  rt  and  rtl 
at  B  and  C  will  cut  the  curves  A4D,  and  A,,D,,. 


Fig.  6.  Fig.  7. 

From  Fig.  6  we  have 

PC  =  h  =  d  sin  p  -  r,,  sin  (180  -  (p  +  9)), 
BF  =  r,  -  [d  cos  p  +  r,,  cos  (180  -  (p  +  0))] 
d  sin  p  —  r,,  sin  (180  —  (p  +  0)) 


=  tan  SF 


BF 

and  since 


''  r,  -  [d  cos  p  +  rtl  cos  (180-  (p  + 


(1) 


(2) 


32  TRANSITION    CURVE 

FC  h  dBmp-r,,sm(lSO-(p+0)) 

BC-r,-(r,,  +  f)-  r/_(r//  +  /) 

D,,CD4  =  S  =  180  -  (P  +  0),  *  -  S  =  u  =  Z^CA,,  ;    (4) 

whence  ry/w  =  Ay/D/x, 

in  which  w  is  the  arc  of  a  circle  with  radius  =  1.  (5) 

If  S  >  *•,  then  the  point  Ayy  lies  between  Dy/  and  D3 
and  the  distance  A/yZ)3  is  measured  from  D3  towards 
Dit  to  A/y. 

If  SE'  >  S,  the  point  Ayy  lies  beyond  J>/y  and  the  dis- 
tance AyyDyy  is  measured  from  Z)yy  to  A/y  whence  Ayy  is 
established.  On  a  perpendicular  to  a  tangent  at  Ay/  lay 
off  /y  and  establish  Ar 

When  /  is  small,  the  direction  of  the  radial  line  can 
be  estimated  near  enough.  The  method  of  fixing  By 
and  J5/y,  in  Fig.  2,  has  already  been  indicated.  If  ry  =  oc, 
Z>yAy  is  tangent.  Fig.  7  applies. 

EXAMPLE  11.  r//  =  1000;  r/y  =  600;  d  =  300;  P  =  70°;  «/  - 
96°;  /  =  50/: 

First,  to  find  h  =  d  sin  P  -  r//  sin  (180  -  (p  4-  «/))  =  300  X 
.93969  -  60  X  .24192  =  136.75  =  FC. 

_  h  136.75 

Second,  to  findsm  *  -      r 


K,00  -  (600  +  50) 


Third,   to  find  180  -  [70°  +  96°]  =  14°  =  2;   *  -  2  =  w  =  23° 
o 

-  14°  =  9°  <a  reduced  to  arc  =  z=~SS  =  -1571- 
o  <  .^y 

rw  =  600  X  .1571  =  94.26  =  ^1//Z>//,  and  since  *  >  2,  ^//Z),/  is 
measured  from  £>//  towards  Z>3. 

If  2  =  D4/CD3,  then  2  >  *  and  yl//Z>3  would  be  measured  from 
Z>3  towards  D//  to  establish  point  An\  A/  is  on  a  perpendicular  to  a 
tangent  to  the  curve  with  radius  r//  at  A//. 


THE    SPIRAL 


33 


PROBLEM  III. 
COMPOUND   CURVES. 

§  14.  Given  two  circular  curves  with  radii  r,  and  r3,  respec- 
tively, whose  centers  are  apart  a  distance  AC  =  b  =  rx  —  (r3  +  ///), 
and  which  are  separated  from  each  other  a  distance  Z>/D//  =  ///. 
It  is  desired  to  introduce  between  them  a  third  curve  with  radius  r//, 
less  than  r/  and  greater  than  r3  and  to  join  the  curve  with  radius  r// 
with  those  having  radii  r,  and  r3  by  means  of  transition  curves: 


Fig.  8. 

By   the    figure   AD,  =  A  A,  =  AB,  =  r, 
=  b  +  r3  +  ///;  whence 

AC  -  b  -  r,  -  (r8  +  /„);  (1) 

AS   =  c  =  r,  -  (r,,  +  f,);  (2) 

BC  =  a  =  r,/  -  (rs  -f  /3);  (3) 
a,  6  and  c  form  the  sides  of  a  triangle  A5C  in  which 

_  L/2  /  i         i^\         _  W  /j^         i  \ 
f/  ~  2T  U,  "  rj  ;  f3  ~  24   Vr3       r^j  :  (4) 


34  TRANSITION    CURVE 


Any  angle  A  may  be  found  by  the  formulae, 

VerA  =  2(s~b)^(s~  C)    in  which  a  =  Ko+b  +  c);  (6) 


Sin£  =       sin  A;  (7) 

Sin  C  =  I  sin  A.  (8) 

Reducing  each  of  the  above  angles  to  an  arc  as  indicated  in  another 
part  of  this  book,  we  have 

A,D,  =  r,A\  AJ),,  =  r3C;  A,,A*  =  r,,  (180  -  £);  (9) 


_i4_   iL_JL). 

r/// 


the  arc 

_ 

2r3        2r3  Vr3 

L3          P    /i          i\ 

P3  =    --  ~~    =    -  I  —    —    -  I  » 
2t//          2f//  \T3          r/// 

in  the  same  way, 


/ 

2r/       2r/\r//       r// 
A/B/  =  r/^/;  A//  B//  =  r//^//  ;  A3B3  =  r/,/33; 

A4B4  =  r3^4;  (14) 

B^Bs  =  r,,  t  ;  t  i  =  (180  -  B)  -  (£//  +  j33);  (15) 

B/D,  =  r/A  +  r/j8/  =  r/  (A  +  £/);  (16) 

D^B4  =  r3  (C  +  /34).  (17) 

EXAMPLE  12.  Given  r/  =  2865;  r//  =  1146;  r3  =  716.3;  ///  =  20; 
P  =  171900;  L,  =  90;  L3  =  90  .'.  //  =  f3  =  .17  to  join  the  circular 
curves  B^Di  and  B4D/t  by  Prob.  III.  First  finding  a,  b  and  c  by 
(1  -  3)  we  have  by  (6)  ver  A  =  3°  44';  arc  A  =  .6516;  by  (7) 
(180  -  B)  =  18°  48';  arc  (180  -  B)  =  .32811;  by  (8)  C  =  15°  06'; 

*  For    determining    large    central    angles    of    £/,  0//,  £3  and  /3 
the  method  of  §  4  is  to  be  preferred. 
t  See  Appendix  I. 


THE    SPIRAL  35 

arc  C  =  .26054;  AJ>,  =  2865  X  .06516  =  186.68;  AJ>tl  =  716.3  X 

Qrt 

.26054  =  186.77;  A,,A3  =  1146  X  .3287  =  376.69;  ft  =  ft/.: 


.6282;  j33  =  /32  =  2(^46)  =  .03926  ;  /3,  =  2(2865  =  -01571  AtBl 
=A,,Bit  =  ^3^3  =  ^4#4  =  45.0  ;  i  =  18°  48'  -  (2°  15'  +  2°  15') 
=  14°  18'  arc  <-  =  .25017;  BtlBz  =  1146  X  .25017  =  286.7;  B,D,  = 
2865  (.06516 +  .0157)  =  231.67;  B*Dlt  =  716.3  (.26054  +.06282) 
=  231.62. 

If  we  make  //  =  0  and  r/  =  r//,  then  c  =  0,  C  —  0,  ///  and  £//  = 
0,  b  =  a  and  is  coincident  with  it,  A  =  180  —  B  and  the  problem 
reduces  to  uniting  A,/A3  with  Z>//#4  by  means  of  the  transition 
curve.  If  TII  =  r/,  c  —  0,  C  =  0,  in  which  case  //  and  f3  =  0;  a  is 
equal  to  and  coincides  with  b  and  A  =  180  —  B  =  0  equations  for 
vers  and  sines  =  0.  The  problem  reduces  to  uniting  curves  B/D,  and 

B^D//  by  means  of  a  transition  curve  whose  length  Z/2  = 


_  J. 

If  f3  and  //  =  0,  while  r/,  r//  and  r3  retain  their  values,  the  transi- 
tion curves  disappear  and  the  curve  with  radius  r//  compounds  at 
A*  and  A,  with  the  curves  having  radii  r,  and  r3. 

180  -  B  is  the  central  angle  of  AtlA3;  (18) 

C  =  that  of  7)//A4;         4  =  that  of  A,D,.  (19) 

If  the  compound  circular  curve,  Prob.  Ill,  be  of  two  centers 
with  radii  r//  and  r/,  then  Eq.  12  and  13  give  the  value  of  /3/  and  £// 
if  the  central  angles  be  not  exceeding  10  or  15  degrees  and  the 
radii  do  not  exceed  955  feet;  otherwise 

Let  o-n  —  o-i  =  $u  +  ft/  =  7  and  r//  and  r/  be  known ;  then,  Eq.  (6) 
page  5 

P==    2^T_^>  (20) 

iv72  ~^2 
and  by  (6) 


and  /  computed  by  Eq.  (37)  then  by  §  4 

£//  and  £/  can  be  obtained,  whence  r//j3//  =  the  length  of  one 
branch  of  the  curve  and  r/P,  the  other.  Then  to  find  the  B.  C., 
C,  C.  and  E.  C.  of  the  compound  circular  curve,  having 

a//  -  a,  =  £„  +  ft,  =  7.  (21) 


36  TRANSITION   CURVE 


r//  tan  $  $,i  —  the  semi-tangent  for  r///3x/  and  n  tan  \  £,  =  the 
semi-tangent  for  r//3/ 

whence  T//  +  T  =  r//  tan  £  £,/  4-  r/  tan  £  B/  =  the  hypotenuse  of 
a  triangle  in  which  one  side  and  three  angles  are  known,  viz.: 
Tt,  +  I1/;  0//  and  ft/  and  7  =  ft//  4-  /3,  and  for  any  side  opposite  £// 
we  have 

b,  =  (IT,,  +  T)  sin  j8//. 

By  a  similar  process  we  may  find  6,/  whence  T,/  4-  &//  =  the 
semi-tangent  adjacent  ft//  and  I7//  +  &/  =  the  semi-tangent  adjacent 
/3,  which  fixes  the  B.  C.  and  E.  C.  and  r/jS,  the  distance  on  the 
curve  to  C.  C.  The  distance  r//3,  assumed  to  be  measured  on  the 
curve.* 

If  it  be  desired  to  introduce  transition  curve  at  the  extremities 
of  this  compound  curve,  the  formulas  of  Problem  I  are  applicable 
by  writing 

T,,  4  b//  and  T  +  b,  for  T. 

When  the  rate  of  curvature  of  the  branches  of  compound  curve 
differ  from  each  other  by  less  than  two  degrees  per  station  of  100 
feet  the  transition  curves  may  be  omitted  at  the  compound  points 
and  introduced  only  at  the  ends  where  the  curve  merges  into  the 
tangents. 

The  semi-tangent  to  a  compound  curve  of  more  than  two  cen- 
ters may  be  computed  by  latitudes  and  departures. 


CASE  2. 

If  it  be  desired  to  introduce  a  third  circular  curve,  exterior  to 
and  joining  the  other  two  fixed  circular  curves,  by  means  of  spiral 
arcs,  all  turning  in  the  same  direction,  then  the  conditions  are 
indicated  in  Fig.  (8a)  and  the  solution  similar  to  the  first  case.  P 
has  fixed  value  from  which  //,  /a,  L/  =  B/B//  and  £3  ==  B^B^  are 
computed.  If,  however,  we  wish  to  displace  the  third  circular 
curve  by  spiral  arcs,  tangent  each  other  at  some  point  O,  having  a 
maximum  radius  of  curvature  =  r//,  then  the  first  approximate 
solution  will  be  obtained  as  follows  :  By  Fig.  (8a)  in  the  triangle 
ABC  let  the  side  AB  =  r,,  -  r/,  BC  =  r,,  -  rs  and  AC  =  r,  +  r3 
±  f,  the  ±  sign  depending  on  whether  AC  is  greater  or  less  than 
r/  +  rs.  Solving  for  angle  B,  we  have,  approximately,  the  central 
angle  of  arc  AnA%  and  r//B  —  arc  A//A&  Let  LO/  and  LOS  —  the 


*  The  chord  of  a  10°  curve  with  radius  =  573  ft.  =  49.99  ft- 
for  an  arc  of  50  feet. 


THE    SPIRAL 


37 


length  of  the  spiral  arcs  computed  from  this  data.      Since  £„:  /3a  = 
r3  :  r,  then  ^j-  =  r//|3//  =  A//O/,    ^  =  r//03  =  AgO//. 

A^On  +  A/,O,  <  A,,A3,  generally,  by  an  amount  O/O//.      Make 

°'°  -  and  °"°  =  °'°"  ~  °'°  then  2  +  °'°     =  L' 


Fig.  8a. 

2  (~  +  O//O/J  =  1/3  nearly,  compute  f/  and  /3  and  make  the  sides 
of  the  triangle,  AB  and  AC,  respectively  =  r,,  —  (r/  +  f/)  and 
TH  —  (r3  +  fs)  and  compute  OA,,  and  OA3  repeatedly  till  =  xf,  and 
x/s,  or  O/O//  =  0  (or  iota  =  0).  If  we  make  r//  infinitely  great,  then 
A uA  3  will  be  a  straight  line  and  O  the  common  origin  of  the^spirals 
in  which  r/  +  r3  ±  ///  =  zw/  +  xmz  and  we  would  have  a  special 
value  of  P  for  these  particular  curves.  For  a  compound  circular 
curve  f/  and  fs  =  0  as  in  the  first  part  of  this  case. 


PROBLEM  IV. 

§  15.  Given  two  curves  turning  in  the  same  direc- 
tion, whose  centers  are  a  distance  apart  =  D,D,,  and 
whose  radii  are  r,  and  rfl.  To  fix  the  position  of  a  tan- 
gent A.tAi4l  and  connect  it  with  the  circular  curves  AttCt 
and  A4C,,  by  means  of  transition  curves  having  a  fixed 
value  of  P  (Fig.  9).  Let  C,C/t  be  a  line  joining  any  two 
points  Cf  and  C/y  of  the  circular  curves  with  radii  rt 


38 


TRANSITION    CURVE 


and  r/y;  measure  the  angles  T>4CtCin  C^C^D^  and  line 
<?,<?„.  By  traverse  we  find  the  angles  C,D,D,,,  C^D^D, 
and  distance  D^D,,  between  the  centers  of  the  curves  rt 


and  ryy.  If  from  Dy  and  D/y  we  let  fall  perpendiculars  to  an 
imaginary  tangent  passing  through  the  origins  of  the 
transition  curves,  by  the  conditions  of  the  problem  the 
length  of  these  perpendiculars  will  be: 

D,A,  =  ry  +  mt  and  DyyA/yy  =  ryy  +  ^//, 
and  any  distance  D4K,  =  (ry  +  mt)  —  (ry/  +  my/),          (1) 
and  if  we  denote  the  distance  D,D,,  by  D4)  then 


IQ  n  _ 
= 


!,)-  (r/y  +  m/x) 
~ 


(2) 


THE    SPIRAL 


39 


C,D,A/f  =  difference  of  C,D,D,,  and  0.  CtDtAtl  reduced 
to  arc  and  multiplied  by  rt  =  the  distance  CfAti  to 
establish  Alt\  from  Alt  lay  off  my  normal  to  the  circle  and 
establish  Ar  In  the  same  way  C,,D,,A4  =  the  differ- 
ence between  180°  -  0  and  £„/>„!>,.  C^Df/A4  mul- 
tiplied by  rlt  =  arc  distance  C/7A4  from  C/x  to  A4  to 
establish  A4,  from  wrhich  lay  off  mtl  and  establish  AUI\ 
then  a  line  through  AtAtll  is  the  required  tangent. 

The  distance  Af)t  to  the  origin  0,  =  xt  —  r,  sin  a, 
and  the  distance  AI4IOI{  =  £„  —  rtl  sin  a/x ;  whence  the 
distance 

0,0,,  ^D.D,,  sin  e  -  [(*„  -r/7  sin  ay/)  +  (X/  -r,  sin  a,)].  (3) 
If  we  wish  the  distance  OtOlt  =  zero,  make 
D/D//  =  [(x/y  -  r/y  sin  av)  +  (xx  -  r,  sin  a,)]  cose  0.     (4) 

This  gives  the  shortest  distance  possible  between  the 
centers  of  the  circular  curves  when  transition  curves 
are  introduced.  The  transition  curves  may  have  differ- 
ent values  of  P  provided  Aflt  +  AtliOtl  =  or  <  than 
AtAtil.  If  the  curves  are  reversed,  D,Kit  —  (r/  +  my) 
+  (r/y  4-  w/y)  and  we  find  the  value  of  ^  by  completing 
the  traverse  CtCtiiD,tiDtCt. 

EXAMPLE  13.  Given  Z)/C/C/y=  85°;  C/C//D/  =  100;  C/C//  -  300: 
r/  =  1146;  r//  =  573.  First,  to  find  D/Z>//  =  D4.  By  traverse  with 
Z)/  as  initial  point. 


From 

Course 

Distance 

Lat.  JV.+ 

Lat.S- 

Dep  E  + 

DepTF- 

To 

D, 

South 

1146.0 

1146.0 

c, 

c, 

N85°E 

300.0 

26.1 

298.9 

Cn 

c* 

N  5°  E 

573.0 

570  8 

49  9 

Dlt 

+  5i>6.y  -  1146.0  +  348.8 

596.9 
-  549.1  +348.8 


40  TRANSITION    CURVE 


Z>,Z>//  =  D4  =  \/(549.1)2  +  (348.8)2  =  650.5 

m,  =  .13     m//  =  1.05     r,  +  m,  =  1146.13     r,,  +  m//  =  574.05 

C08  s  _  frv  +  m,)  -  (r,  +  m,,)  _  W|0|  _  0_87944  =  OQS  2go  25, 

4  635-22  =  tan  32°  25/ 


32°  25'  -  28°  25'  =  4°     .'.     D,A,  =  r,  +  m,  bears  S  4°  E. 
At,  A,  =  m/  =  .13'.     By  the  table  C//Z>//  bears  ^V  5°  ^. 
Di,A,n  =  r//  +  w//;    parallel  to  D,Ar,  bears  /S  4°  E.     C//D//J.///  =  9° 
^.4A///  =  1.05     A,A,lt  =  T>,iK,  =  Z)4sin  B 

=  A±Atll  =  650.5  X  .4759  =  309.5 

O,O//  =  D/Z>/  sin  Q  -  \(x,,  -  r,,  sin  a,,)  +  (a:/  -  r,  sin  a)]. 
Let  a://  =  119.90;  a/7  =  6°     a;/  =  60';  a,  =  1°30/ 

O.On  =  309.5  -  [59.95  +  30]  =  309.5  -  89.95  =  219.55 
to  make  O/O//  =  0 


M///  =  89.95  cose  8,  or  Z>/Z>///  =  89.95  X  6.4398  =  579.25. 


PROBLEM  V. 
OLD  TRACK. 

§  1 6.  To  introduce  the  transition  curve  in  align- 
ment where  circular  curves  have  been  run. 

In  Fig.  10,  suppose  ABC,  a  simple  circular  curve,  to 
have  been  run  tangent  to  the  line  OGtl  at  A  with  radius 
rt  =  DjA  =  D,B;  it  is  desired  to  introduce  a  spiral  whose 
greatest  curvature  has  a  radius  ru  =  D^A,,  =  &,,&„  = 
DltB  <  r,.  From  the  tables  or  by  computation  we  have 
m  depending  on  the  value  of  ru  and  Z//7;  with  given 
values  of  m,  rtl  and  r,  we  have  from  the  figure 

AD,  =  A,A,/  +  AiPu  +  D«D<  cos  D«D<K'> 
or  rf  =ml  +  rtl  +  (r,  —  r/y)  cos  0  /. 

^  =  r,  -  rtt  -  (r,  ~  r,,)  cos  0 
m  =  r/  -  r,,  (1  -  cos  0)  =  (r,  -  ry/)  ver  0 

ver  0  ==  -    — —  ;  r,  <f>  =  AB ;  ry/  0  =  Ay/B 


THE   SPIKAL 


41 


AB  =  the  distance  to  measure  from  A  =  PC  to  locate 
B;  BAlt  the  distance  to  measure  from  B  on  BAU  to 
locate  Alt.  From  m  and  ru  we  have 


0 


4, 
Fig.   10. 

2ry/ay/  =  L  =  \/24mr,,,  squaring,  4r//2a//2  = 
or  2      6m  /6m  „         L 

"fW;  ""        V^;r"a//  =      "    "=2 
r//  (0,,  -  <*„)  =  BB//;  (r,  -  ry/)  sin  ^  =  AyA. 

The  rate  of  curvature  of  ry/  should  not  be  more  than 
from  1°  to  2°  greater  than  that  of  r,  (when  possible)  for 
curves  of  a  curvature  less  than  10°;  2°  to  3°  difference 
for  10°  to  15°  rate  of  curvature;  3°  to  5°  difference  for 
15°  to  20°  rate  of  curvature. 


EXAMPLE  14.     Given  r/  =  1146,  r,,  —  955,  m 
a,,,  AB,  BB,t  and  L 

m  142 


1146  -  955 


1.52,  to  find  <£ ; 
=0.00743=  ver6°59/ ;  arc  #  =  .12188 


42  TRANSITION    CUKVE 

r/0  =  .1219  X  1146  =  AB,  r//0  =  1219  X  955  =  115.8  =  A,,B 


arc  «„  =  .  _  .0944  „„  _  5o  24, 

///  yoo 

/  (0  -  a)  =  955  (.1216  -  .0944)  =  25.7  =  B,,B 
L  =  2/v/a,,  =  2  (955  X  .0944)  =  180.2 


EXPLANATION  OF  THE  TABLES. 

,.  '   BY  RECTANGULAR  CO-ORDINATES. 

§  17.  Tables  1  to  7  give  values  for  laying  out  the 
transition  curve  by  the  method  of  rectangular  co-ordi- 
nates. They  are  equally  applicable  for  uniting  a  tangent 
with  a  circular  curve,  or  curves  having  different  radii, 
by  means  of  the  transition  curve.  Ltl  and  Lt  may  be 
taken  separately  from  the  same  column,  as  also  may  a/y 
and  a,,  and  their  difference  will  be  the  value  of  L  and  <£ 
for  the  length  and  central  angle  respectively.  The  sev- 
eral ordinates,  x,  y,  xf,  yf,  are  laid  off  from  J3,  or  Elt  as 
origin  (Fig.  2)  with  arc  BtA,  or  J5/yAy/  as  axis,  —  the 
same  as  if  B/  or  Blt  were  written  for  A  in  Fig.  3,  and  the 
successive  stations  were  B,  C,  D,  etc.,  and  B,,  C4,  D/} 
etc.,  successive  points  x,  xn  xlti,  etc.,  with  corresponding 
y>  y<)  2///>  etc.,  values  normal  to  the  curved  axis  A4B, 
AuBtl  in  the  same  manner  as  if  A4B,  were  tangent. 
P  and  v  are  taken  of  such  values  as  to  avoid  introducing 
fractions  in  L.  L  is  supposed  to  be  measured  on  the 
curve,  but  since  the  chords  are  generally  quite  short, 
the  sum  of  their  lengths  is  but  little  less  than  that 
of  the  curve,  hence  no  allowance  is  made  for  the  length 
of  the  curve  being  in  excess  of  the  sum  of  the  lengths  of 
the  chords. 

EXAMPLE  1.  Given  L  =  120;  r,  =  1432.5;  r//  =  716.3  (Table 
II),  then  «//  -  a/  =  9°  36'  -  2°  24'  =  7°  12'  =  <j>  with  E  as  origin. 
At  the  end  of  the  first  chord  length  from  E  towards  /  we  have 
x  =  30',  y  =  .03  =  co-ordinates  for  F;  x,  =  60,  y,  =  .21  =  co- 
ordinates for  G\  Xn  =  90,  y//  =  .70  =  co-ordinates  for  H\  x$  = 
119.98,  2/3  =  1.67  =  co-ordinates  for  /;  xf  =  60;  yf  =  .21;  /  =  .42. 


THE   SPIRAL  43 


The  method  of  laying  out  these  co-ordinates  is  shown  in  Fig.  4, 
in  which  the  origin  A  corresponds  to  the  point  E  in  this  example, 
and  AD i  becomes  a  curved  axis  with  a  radius  of  1432,5. 

If  the  transition  curve  were  laid  off  from  BuAti,  Fig.  2  as  axis 
and  Bu  as  origin,  the  above  values  x,  etc.,  and  y,  etc.,  would  be 
just  the  same  except  they  would  be  laid  off  from  the  convex  side  of 
AnBii  instead  of  from  the  concave  side,  as  was  the  case  with  A/B/ 
as  axis.  If  the  curvature  of  the  circular  curve  is  of  fractional 
degree  the  value  of  /  and  the  last  values  of  x,  y  and;  a  will  have 
to  be  computed  by  the  formulae  at  the  head  of  the  respective 
columns  in  Spiral  Table  I. 


BY   DEFLECTION. 
(Or  Polar  Co-ordinates.) 

§  1 8.  Given  a  tangent  at  any  point  of  a  transition 
curve  as  D  to  locate  any  other  points  as  A,  B,  C,  E,  F, 
G  and  H.  As  in  the  case  of  the  Tables  for  .rectangular 
co-ordinates  they  are  equally  applicable  for  locating  the 
points  of  the  transition  curve  uniting  a  tangent  to  the 
circular  curve  or  circular  curves  of  different  radii  with 
each  other.  (Fig.  3.) 

EXAMPLE  2.  Let  L,,  -  L,  =  L  =  120  be  the  length  of  the 
transition  curve;  r/  =  1910;  r//  =  716.3  the  radii  of  the  circular 
curves  to  be  united  by  L  by  deflections  from  a  tangent  at  Z>,  where 
the  curvature  corresponds  to  r/.  The  tangent  will  be  common  to 
the  circle  and  transition  curve  whose  rate  of  curvature  "  D"  (Table 
II)  =  3°.  DE,  EF,  FG,  etc.,  being  chords  of  the  transition  curve 
each  =  30  feet.  The  tangent  at  D  is  a  tangent  to  the  circular  arc 
with  rt  =  1910.  Then  by  the  formula  w  =  L[D&  +  N  X  0°00.1'1 
in  which,  if  we  write  L  =  30,  60,  90  and  120;  and  N  =  1,2,  3  and  4 
successively,  D  =  3,  A  =  0.3',  then  the  deflection  from  tangent  at 

D  to  locate  E  is  o>  =  30  [3  X  .3'  +  1  X  0°  OO.t']  =*  0°  30' 
D  to  locate  F  is  <w  =  60  [3  X  .3'  +  2  X  0°  OO.l'l  =  1°06' 
D  to  locate  G  is  a  =  90  [3  X  .3'  -f  3  X  0°  OO.l'l  =  1°  48' 
D  to  locate  H  is  <»  =  120  [3  X  .3'  -4-  4  X  0°  OO.l'l  =  2°  36' 


44  TRANSITION    CURVE 

with  measurement  from  D  to  E  thence  E  to  F,  etc.  If  we  wish  to 
locate  the  points  C,  B  and  A  from  a  tangent  at  D,  then  the  deflec- 
tion for  any  point  C,  for  example,  =  the  deflection  for  30  feet  for  a 
3-degree  curve  minus  the  deflection  for  the  transition  curve  from 
A  to  B  by  formula  (61). 

hi  =  L  [DA  -  N  X  0°  OO.l'l,  whence  from 
D  to  locate  C  is  aj  =  30  [3  X  .3'  -  1   X  0°  00.1']  =  0°  24' 
D  to  locate  B  is  w  =  60  [3  X  .3'  -  2  X  0°  00.1']  =  0°  42' 
D  to  locate  A  is  w  =  90  [3  X  .3'  -  3  X  0°  00.1']  =  0°  54'. 

If  one  of  the  chords  be  fractional  and  the  change  of  curvature 
per  chord  be  fractional  also. 

Let  j,  =  15'  -  i,  C  =  1°  30'  =  U 
then  by  (61 ) 

at  =  L[Z>A±(  A'  +  J;)  C  X0°00.1']. 

To  locate  any  point,  as  G  +15  from  tangent  at  D,  then 
at  =  105  [3  X  .3  4-  (3  +  i)  X  H  X  0°  00.1']  =  2°  29'. 

To  locate  any  point,  as  A  +  15'  from  tangent  at  D,  then 
<o  =  75  [3  X  .3'  -  (2  +  |)  H  X  0°  00.1']  =  0°  39'. 

If  we  have  run  the  curve  from  A  to  D  and  changed  the  instrument 
to  D  in  order  to  place  the  line  of  sight  tangent  to  jD.take  a  back- 
sight on  A  and  deflect  0°  54'  and  we  have  a  tangent  at  Z>.  To 
facilitate  the  use  of  the  tables  it  is  best  to  set  the  vernier  at  0°  54' 
and  set  the  telescope  on  line  AD,  turn  the  vernier  to  O  and  con- 
tinue deflection  as  tabulated,  reading  downward  from  D,  locating 
the  points  E,  F,  G,  etc.  If  the  curve  is  being  run  from  D 
towards  A  then  set  the  vernier  at  the  angle  indicated  for  any  angle 
G,  when  backsight  is  on  G  from  D  deflect  from  zero  and  continue 
to  deflect  the  angles  tabulated  in  succession,  reading  up  the  column 
from  D  to  locate  C,  B  and  A.  The  degree  of  the  curvature  at  the 
instrument  point  controls  the  deflections  either  way.  The  above 
explanation  enables  us  to  run  the  transition  curve  from  the  point 
of  greatest  radii  to  that  of  its  least  radius,  and  vice  versa. 

If  we  take  the  curvature  "Z>"  at  the  position  of  the  instrument 
as  the  basis  of  calculation,  then  equation  (61)  can  be  applied 
directly  to  get  on  tangent  at  the  position  of  instrument  (after  back- 
sight on  the  last  instrument  point),  using  the  —  sign  for  running 
towards  G  and  the  4-  sign  for  running  towards  .4. 


THE    SPIRAL  45 

TABLE  OF  CIRCULAR  ARCS 


Length  of  Circular  Arcs  at  Radius  =  i 

Decimals  of  a  Degree 

Sec. 

Length 

Min. 

Length 

Deg. 

Length 

Min. 

Decimal 

.Sec. 

Decimal 

1 

.000005 

1 

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APPENDIX  I 


PROBLEM. 

Given:  The  altitude  of  two  circumpolar  stars,  nearly  opposite 
each  other,  the  local  time  of  Meridian  passage  of  either  star,  their 
difference  in  time  of  Meridian  passage  and  the  polar  distance  of 
each  star, 

TO    FIND   THE  LATITUDE,  THE    LOCAL    TIME  AND   THE 
MERIDIAN  OF  THE  PLACE. 


Fig.    77. 

For  convenience,  let  Polaris  and  Alioth  be  the  stars  observed; 
O  the  position  of  the  observer  and  center  of  the  celestial  sphere. 
ONEW  be  the  north  half  of  that  portion  of  the  celestial  sphere 
above  the  horizon  WNE,  with  A  as  zenith. 

Let  GC  =  c  or  G//C//  =  c,,  be  the  altitude  of  Alioth  at  the  time 
of  observation  and  BS  =  g  or  £//*£/.•  =  g//  the  altitude  of  Polaris  * 
35  minutes  later;  then  the  plane  passing  through  the  points  S,  G 
and  O  will  pass  through  the  pole  P  and  cut  the  celestial  sphere  in 
the  line  r  +  r,  SG  or  £//G//,  r  t  and  r/  being  the  polar  distance 
respectively  of  Polaris  and  Alioth  and  AP  =  s  the  co-latitude  of 
the  place  O.  Then  the  plane  ASO  or  AS//O  will  cut  the  side  of  the 


*  At  this  date. 

t  The  position  of  these  stars  have  a  slight  annual  change  with 
reference  to  the  pole,  which  must  be  taken  into  account  from  year 
to  year. 

59 


60  APPENDIX 

spherical  triangle  AS  =  AB  -  BS  =  (90  -  0)  or  A8lt  =•  ABlt  - 
£//£,/  from  the  celestial  sphere.  Similarly,  AGO  or  AG,,O  will  cut 
(90  -  c)  or  (90  -  c,/).  The  sides  /,  r  +  r/  and  (90  -  c)  or  (90 
—  c//)  form  the  spherical  triangle  ASG  or  J.*S//G//,  in  which  the 
sides  are  known;  from  which  we  find  the  angle  S  or  £//  by  the 
spherical  formula: 

(cos  90  -  c)  -  cos  (90  -  g)  cos  (r  +  r) 

COS  AoOr   =  COS  o    = ; ,nn       — r — r — 7 r 1 

sin  (90  -  g}  sin  (r  +  n) 
but  cos  (90—  c)  =  sin  c  and  cos  (90  —  g)  =  sin  0,  hence: 

„       sin  c  —  sin  g  cos  (r  4-  r/) 
cos  0  sin  (r  4-  r/) 

The  angle  £  is  common  to  the  two  triangles  ASG  and  ASP. 

We  now  have  r,  S  and  (90  —  g)  to  determine  A  —  the  Azimuth, 
Z  =  the  Latitude  and  P  =  the  Hour  Ang  e  (expressed  in  degrees). 

cos  (90  -  I)  =  cos  (90  -  g}  cos  r  ±  sin  (90°  -  g)  sin  r  cos  S, 
or  sin  1  =  sin  g  cos  i  ±  cos  g  sin  r  cos  S,  (a) 

using  —  sign  when  S  >  90;  and  since 

sin  (90  -  Z):  sin  (90  —  0)::  sin  S:  sin  P, 
or  cos  Z:  cos  g::  sin  /S:  sin  P 

.    _        cos  g  sin  S  /tv 

sin  P  = ^— : (b) 

cos  1 

P  represents  the  angle  the  plane  OSP  makes  with  the  plane 
OPA  at  the  time  of  observation  on  S  and  P°  X  4  the  Hour  Angle, 
expressed  in  minutes  of  sidereal  time,  from  the  plane  ONPA.  Also 
in  the  triangle  ASP  we  have 

sin  r  sin  P         sin  r  sin  P  ,  N 

sin  A  =  - — ; •  =  —      •  (c) 

sin  (90  —  g)  cos  g 

in  which  A  is  the  angle  between  the  planes  ABO  and  ANO,  which 
is  the  Azimuth  Angle  for  the  Meridian  Plane  ANO. 

A  similar  solution  applies  to  the  triangles  AS//P  a.ndAS«G/i 
Equations  (a),  (b)  and  (c)  solve  the  problem. 

When  the  altitude  of  Alioth  is  much  in  excess  of  that  of  Polaris, 
the  observations  are  not  easily  made  in  higher  latitudes  with  the 
ordinary  engineer's  transit,  unless  equipped  with  a  prismatic  eye- 
piece. 

Sidereal  hours  X  .9972696  =  mean  solar  hours. 


MERIDIAN  61 


EXAMPLE.     Given  g  =  40;  c  =  20;  r  =  33°  30'  to  find  the  lati- 
tude I.     We  first  find  the  angle  S  by  the  formula, 

c  =  sin  c  -  sin  g  cos  (r  +  r,  )  =  .3292  -  .64279  X  .82181 
cos  g  sin  (r  +  r)  .76604  X  .56976 


To  find  the  latitude  we  have  (a) 

sin  I  =  sin  fir  cos  r  ±  cos  g  sin  r  cos  S 
using  the  —  sign  (since  S  >  90°), 

sin  I  =  .64279  .9998  -  .76604  X  .0215  X  42667  =  .642661  -  00702 
=  .63564  =  sin  39°  28',  whence  the  Latitude  39°  28'. 

To  find  the  local  time  we  have  (6) 
sin  P  i.  '°M|iHjS  _  -76604^90445  =  ^^  _  ^  ^  ^  p  ^  fm 

=  63°  49'  X  4  =  4h  15m  =  255  sidereal  minutes;  255  X  .99727  =  254 
mean  solar  minutes,  or  P  =  4"  14m. 

The  time  of  observation  was  at  1  o'clock  A.M.,  Dec.  1st,  1900, 
i.e.  13h  -  4h  14m  =  8h  46m  the  time  of  meridian  passage  by  the 
clock. 

The  local  time  of  meridian  passage  was  8h  40m  i.e.  8h  46m  - 
8h  4Qm  =  Qh  Q6m  {e  the  clock  was  6m  fast> 

To  find  the  Azimuth  Angle  SAP  =  BON  =  A,  we  have  (c) 


sin  r  sin  P        .0215  X  .7974        no_1c     , 

sin  A  =  —  =    _Qgr. , =  .02518  whence  ^4=1°  27' 

cos  g        .76604 


SAN  FRANCISCO,  1898. 


62 


APPENDIX 


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MISCELLANEOUS   PROBLEMS,    SUPPLEMENTS     63 


METHOD  OF  COMPUTING  RADIUS. 

The  usual   method   of  computing   the  radius  of   curvature  of 
circular  curves  used  in  railroad  location,  is  by  the  formula 


in  which  c  represents  a  chord  of  50  feet  in  length,  and  D  is  the  degree 
of  the  curve  or  central  angle  for  a  chord  of  100  feet.  By  this  method 
the  radius  does  not  vary  inversely  as  the  degree  of  the  curve;  while 
in  the  method  of  this  book  it  is  assumed  to  do  so,  and  r  is  assumed 
to  equal  the  radius  of  a  one-degree  curve  divided  by  the  degree  of 
the  proposed  curve,  or 

r  =  5-f?.  (2) 

These  formulae  give  the  same  result  for  a  five-degree  curve  for 
50-foot  chords,  thus: 


from  which  we  see,  by  the  method  of  this  book,  the  radius  of  a  five- 
degree  curve  is  the  same  computed  by  either  formula.  For  less 
than  5°  rate  of  curvature  r  eq.  (1)  <  r  eq.  (2)  and  for  a  rate  of  curv- 
ature greater  than  5°  r  eq.  (1)  >  r  eq.  (2)  for  50-foot  stations. 

Since  the  length  of  the  chords  is  usually  less  than  50  feet,  in  the 
spiral  the  radius  of  curvature  is  assumed  to  vary  inversely  as  the 
degree  of  the  curve,  and  the  same  radius  may  be  used  for  the  main 
curve,  by  making  its  chords  less  than  50  feet,  the  proper  length  to 
be  determined,  by  computation,  and  tabulated  for  field  use.  It 
is  scarcely  worth  notice  (=  .002  per  degree  of  curve  less  than  50 
feet  for  each  degree  in  rate  of  curvature  per  100  feet  of  arc). 

TO  DETERMINE  THE  NUMERICAL  CONSTANT  IN  FORMULA  (60): 

Place  0°00.1'  =  k,  then  we  have  8'  =  LNCk 
or  k 


We  find  6'  by  dividing  eq.  (21)  by  eq.  (26),  whence, 
tan  5 


y        6r\         56r2/                                6r  V      56r2  + ) 
'-  *  -  ; F7T  +.  <>r  ^n  «  =   =- /- 


zr 

~ 


64  APPENDIX 

5  should  be  reduced  to  minutes  (')  before  entering  the  formula 
for  finding  the  value  of  k. 

The  above  value  of  k  is  for  running  in  a  direction  from  A  towards 
B,  Fig.  3,  or  D  towards  E,  etc. 

For  running  in  a  direction  from  G  towards  F,  or  D  towards  C: 
Having  a'  (minutes),  the  total  number  of  minutes  in  the  central 
angle  of  any  given  length  L  of  the  spiral,  corresponding  to  5'  in  the 
above  equation,  if  we  substitute  (a'  —  8')  in  eq.  (1),  of  this  supple- 
ment, for  S  and  2L  for  Lt  then  calling  the  numerical  coefficient  k', 
we  have 

.          a/  ~  5'« 


for  running  in  a  contrary  direction  from  the  same  points  as  in  eq. 
(1)  of  this  supplement.  k/  should  always  come  out  greater  than  k. 

The  value  of  k,  will  be  found  to  increase  and  k  decrease, 
slightly,  with  a.  For  L  =  200  feet,  or  less,  and  a  =  10°,  or  less, 
we  may  make  k,  =  k  =  0°00.1'. 

It  will  be  seen  by  a  trial  example  that  the  difference  between 
this  method  and  that  of  eq.  (61)  for  a  curve  with  L  =  400  (with 
change  point  at  200  feet)  and  a  =  16°,  is  0°00'03". 

The  difference  increases  with  the  central  angle  a.  The  above 
method,  with  change  points  200  to  300  feet  apart,  is  quite  accu- 
rate and  best  for  preparing  a  set  of  tables,  though  not  so  easily 
applied  in  field  computations  as  eqs.  (60)  or  (61). 

By  assuming  "change  points"  from  100  to  250  feet  apart  the 
values  in  the  deflection  tables  of  this  book  may  be  substituted  for 
the  second  term  in  eq.  (61)  and  the  spiral  extended  indefinitely. 

The  following  table  shows  the  results  of  computations  by  the 
above  formulae. 

The  value  of  the  constant  C  in  each  particular  case  is  determined 

by   ~  ~  C,  in  which  D,  represents  the  degree,  or  rate  of  curvature, 

at  the  point  to  be  located;  this  substituted  in  eq.  (60b)  reduces  it 
to  the  form  of  8'  =  LD,k.  The  above  deduced  value  of  C  should 
always  be  applied  in  eq.  (60)  or  (60b). 

k  is  not  a  constant,  but  for  small  deflection  and  central  angles 
its  successive  differences  are  inappreciable,  but  for  large  central 
angles  these  differences  may  not  be  disregarded. 

If  8'  is  constant 

k  =  LD7'  L  =  W  ;  D'  =  kL 


,  ,         ,        a'  -  8'  a'  ™  _ 

and  for     fc,=     orr.    ;  L  -    .,  n     ;  D,  =  • 


2k,D,    '  "     2k,L 


MISCELLANEOUS   PROBLEMS,    SUPPLEMENTS       65 


^ 

CO       CO 

8  8  i  i 

. 

^ 

41 

O       O       O       O       O       Oi       C3 

r-(         r-         i-l         r-l         t-H         O         O 

fill 

«0 

1 

^                                                    LO       CO 
»C       W                                       <N       CO 

CO 
CO 

^H      cq      <N     co 

« 

00       —        W       00       O       1>       *-H 

CO 

r-*       00       CD        O? 

o°       o         o"1       o 
00       O       CO       CO 

5ft|  H 
§ 

O       »O       W       CO       <N       O       <N 
»O       C^       Oi       <N       CN       cs       ^* 

SO       <N       CO       0       00       1-1 
O       O       O       O       O       i-i 

1111 

O       O       O       O       O       O       O 

o     o     o     o 

,, 

O      t»      *>      9      O      0      9 

"  *   d   §3   S 

O5       CO       00       b» 
CD       CD       OS       b-j 

8  i   §   8  9  „   „ 

t-      t~      Oi      CO 

H 

O       Gi       O5       O5       OS       00       O 

sill 

d 

•^*     o     o     ^     o     ^     o 

N      o°      o°       o*      P      o^      o° 
•H       CO       O       O       ^       Oi 

W       CO       "*       "t 

- 

rt«     oo     M     o     o     ^     oo 

itri 

- 

W       "f       O       00       O       N       ^ 

CO       00       O       N 

»-•         r-l         CQ         W 

66  APPENDIX 

in   which  L  is  in  feet,  Dt   in   degrees  (°),  8'  in  minutes;  m  being 

determined  by  whatever  value  is  assumed  for  L  and  D°  = 

and  substituted  in  eq.  (37)  or  (38). 

SUPPLEMENT  TO  PROBLEM  I. 

The  total  intersection  angle  is 

/  =  a   +  a  +  t. 

t  is  the  angle  due  to  the  circular  portion  of  the  curve  or  B/Btitt 
with  fixed  radius  r//  and  • 

B.DB,,,  =  I  -  (a,  +  a), 

whence,  by  making  the  second  term  in  eq.  (61)  =  0,  we  have: 

a)  —  LZ)A  =  the  deflection  from  a  tangent  at  any  point  of  the 
circular  curve,  as  Bi  or  Btn  to  locate  any  other  point  of  the  circular 
curve  between  Bi  and  Bin  inclusive  of  the  last  point  Bi  or  .B///as 
the  case  may  be. 

SUPPLEMENT  TO  PROBLEM  III. 

t  =  the  arc  of  that  portion  of  the  curve,  between  spirals,  that 
is  circular  and  laid  out  by  the  method  of  circular  curves,  the  deflec- 
tion from  tangent  being  computed  by  eq.  (61),  in  which  the  second 
term,  in  brackets,  is  made  =  0,  whence,  it  becomes  <D  =  LZ)A.  t  will 
have  to  be  reduced  from  arc  to  degrees  and  minutes  to  determine 
the  total  deflection. 

THE  PROCESS  OF  FIELD  WORK. 

Beginning  at  #/,  as  instrument  point,  deflect  from  tangent  OtX 
for  successive  stations  by  formula  (60)  including  Bi  (=  "j.then, 
placing  the  instrument  at  J5/,  with  backsight  on  O/,  deflect 
6'  =  2LNC  X  0°  00.1'  (=  |  a  )  . 

The  line  of  sight  will  then  be  a  tangent  common  to  both  the 
spiral  OtBi  and  the  circular  curve  BtBm  at  £/.  Deflect  thence  by 

formula 

(a  =  LDA 

to  locate  stations  on  the  circular  curve  BiBm  fixing  JS///. 

Set  up  the  instrument  at  B,,t  and,  with  backsight  on  #/,  deflect 
w  —  LZ>A  for  tangent  at  Bm,  thence  locate  points  of  spiral  J5///O/// 
by  applying  formula  (61)  with  —  sign  or  by  (54),  any  deflection 

6'  =  2LNC  X  0°  00.1'. 


MISCELLANEOUS   PROBLEMS,    SUPPLEMENTS      67 


Having  located  O///,  set  up  instrument  at  O///  and  with  back* 
sight  on  Bui  deflect 

5'  =  LNC  X  0°  00.1', 

and  the  line  of  sight  should  coincide  in  direction  and  position  with 
the  tangent  O///X,  or  the  semi-tangent  On  id,. 

Some  prefer  to  run  the  spirals  first,  i.e.,  from  O/  to  Bi  and  from 
Oui  to  B///,  and  the  circular  portion  B<Bm  last,  thus  throwing  the 
closing  error,  if  any,  at  Bi  or  B,n. 

Undoubtedly  this  makes  a  better  adjustment  when  the  points 
Oi  and  On,  have  been  fixed  previously  by  semi-tangents. 

All  change  points  should  be  established  by  double  centers  to 
eliminate  errors  of  adjustment  in  the  transit  instrument. 


TRIGONOMETRIC  TABLES 


^  "c  trigonometric  functions  of  any  angle  intermediate  those  given 
in  the  tables  may  be  found  by  interpolation,  thus: 

What  is  the  natural  tangent  of  12°43'  ? 

From  the  Table,  tan.  12°50'  =      0.22781 
tan.  12°10'  =      0.22475 


Diff.  for  0°10' 

"      "     0°03' 
Add    "    12°4U' 

Hence  for              tan.  12°43'  = 
What  is  the  natural  cotangent 

T3Vr>r»-i   flip  T*aT»1*>      r»r>t      1VO/i/V    — 

0.0030G 
.3 

0.000918*                                   *"+ 
\F          coti 

0.22475                       ^^ 

Si 

0.22567                /       £*  + 
->fl904:^'?      Y 

xfi 

"        Z 

A  AAQAO                     \            ^f 

0       ccs  t    ^Nn 

tl  / 

/x-_  / 

Diff.  for  0°10' 


cot.  12°50'  =      4.38969 

=      0.05973 
.3 


"  0°03'  =    0.017919* 

Subtract  from       cot.  12°40'  =      4.44942 


Hence 


cot.  12°43'  =      4.43150 


To  obtain  functions  not  given  in-  the  tables: 
Vers.  a  —  1— cos.  a  ;  External  Sec.  a  =  1— Sec.  a. 

*The  computation  being  additive  or  subtractive  according  as  the 
function  increases  or  decreases  with  the  increase  of  the  angle  a. 


TRIGONOMETRIC  TABLES 


1 

SINE 

J? 

0' 

icy 

2O' 

30' 

4O> 

50' 

6O/ 

0 

0.00000 

0.00291 

0.00582 

0.00873 

0.01164 

0.01454 

0.01745 

89 

1 

0.01745 

0.02036 

0.02327 

002618 

0.02908 

0.03199 

0.03490 

88 

2 

0.03490 

0.03781 

0.04071 

0.04362 

0.04653 

0.04943 

0.05234 

87 

3 

0.05234 

0.05524 

0.05814 

0.06105 

0.06395 

0.06685 

0.06976 

86 

4 

0.06976 

0.07266 

0.07556 

0.07846 

0.08136 

0.08426 

0.08716 

85 

6 

0.08716 

0.09005 

0.09295 

0.09585 

0.09874 

0.10164 

0.10453 

84 

6 

0.10453 

0.10742 

0.11031 

0.11320 

0.11609 

0.11898 

0.12187 

83 

7 

0.12187 

0.12476 

0.12764 

0.13053 

0.13341 

0.13629 

0.13917 

82 

8 

0.13917 

0.14205 

0.14493 

0.14781 

0.15069 

0.15356 

0.15643 

81 

9 

0.15643 

0.15931 

0.16218 

0.16505 

0.16792 

0.17078 

0.17365 

80 

10 

0.17365 

0.17651 

0  17937 

0.18224 

0.18500 

0.18795 

0.19081 

79 

11 

0.19081 

0.19366 

0.19652 

0.19937 

0.20222 

0120507 

0.20791 

78 

12 

0.20791 

0.21076 

0.21360 

0.21644 

0.21928 

0.22212 

0.22495 

77 

13 

0.22495 

0.22778 

0.23062 

0.23345 

0.23627 

0.23910 

0.24192 

76 

14 

0.24192 

0.24474 

0.24756 

0.25038 

0.25320 

0.25601 

0.25882 

75 

15 

0.25882 

0.26163 

0.26443 

0.26724 

0.27004 

0.27284 

0.27564 

74 

16 

0.27564 

0.27843 

0.28123 

0.28402 

0.28680 

0.28959 

0.29237 

73 

17 

0.29237 

0.29515 

0.29793 

0.30071 

0.30348 

0.30625 

0.30902 

72 

18 

0.30902 

0.31178 

0.81454 

0.31730 

0.32006 

0.32282 

0.32557 

71 

19 

0.32557 

0.32832 

0.33106 

0.33381 

0.33655 

0.33929 

0.34202 

70 

20 

0.34202 

0.34475 

0.34748 

0.35021 

0.35293 

0.35565 

0.358-37 

69 

21 

0.35837 

0.36108 

0.36379 

0.36650 

0.36921 

0.37191 

0.37461 

68 

22 

0.37461 

0.37730 

0.37999 

0.38268 

0.38537 

0.38805 

0.39073 

67 

23 

0.39073 

0.39341 

0.39608 

0.39875 

0.40142 

0.40408 

0.40674 

66 

24 

0.40674 

0.40939 

0.41204 

0.41469 

0.41734 

0.41998 

0.42262 

65 

25 

0.42262 

0.42525 

0.42788 

0.43051 

0.43318 

0.43575 

0.43837 

64 

26 

0.43837 

0.44098 

0.44359 

0.44620 

0.44880 

0.45140 

0,45399 

63 

27 

0.45399 

0.45658 

0.45917 

0.46175 

0.46433 

0.46690 

0.46947 

62 

28 

0.46947 

0.47204 

0.47460 

0.47716 

0.47971 

0.48226 

0.48481 

61 

29 

0.48481 

0.48735 

0.48989 

0.49242 

0.49495 

0.49748 

0.50000 

60 

30 

0.50000 

0.50252 

0.50503 

0.50754 

0.51004 

0.51254 

0.51504 

59 

31 

0.51504 

0.51753 

0.52002 

0.52250 

0.52498 

0.52745 

0.52992 

58 

32 

0.52992 

0.53238 

0.53484 

0.53730 

0.53975 

0.54220 

0.544C4 

57 

33 

0.54464 

0.54708 

0.54951 

0.55197 

0.55436 

0.55678 

0.55919 

56 

34 

0.55919 

0.56160 

0.56401 

0.56641 

0.56880 

0.57119 

0.57358 

55 

85 

0,57358 

0.57596 

0.57833 

0.58070 

0.58307 

0.58543 

0.58779 

54 

36 

0.58779 

0.59014 

0.59248 

0.59482 

0.59716 

0.599i9 

0.60182 

53 

37 

0.60182 

0.60114 

0.60645 

0.60876 

0.61107 

0.61337 

0.61566 

52 

38 

0.61566 

0.61795 

0.62024 

0.62251 

O.G2479 

0.62706 

0.62932 

51 

89 

0.62932 

0.63158 

0.63383 

0.63608 

0.63832 

0.64056 

0.64279 

50 

40 

0.64279 

0.64501 

0.64723 

0.64945 

0.65166 

0.65386 

0.65606 

49 

41 

0.65606 

0.65825 

0.66044 

0.68262 

0.66480 

0.66697 

0.66913 

48 

42 

0.66913 

0.67129 

0.67344 

0.67559 

0.67773 

O.G7987 

0.68200 

47 

43 

0.68200 

0.68412 

0.68624 

0.68835 

0.69046 

0.69256 

0.69466 

46 

44 

0.69466 

0.69675 

0.69883 

0.70091 

0.70298 

0.70505 

0.70711 

45 

e<y 

60' 

40' 

3O' 

2O' 

1O' 

<y 

i 

COSINE 

1 

TRIGONOMETRIC  TABLES 


COSINE 

<y 

icy 

2O* 

SCK 

40* 

60> 

Qfy 

0 

1.00000 

1.00000 

0.99998 

0.99996 

0.99993 

0.99989 

0.9&985 

89 

1 

0.99985 

0.99979 

0.99973 

0.99966 

0.99958 

0.99949 

0,99939 

88 

2 

0.99939 

0.99929 

0.99917 

0.99905 

0.99892 

0.99878 

0.99863 

87 

3 

0.99863 

0.99847 

0.99831 

0.99813 

0.99795 

0.99776 

0.99756 

86 

4 

0.99756 

0.99736 

0.99714 

0.99692 

0.99668 

0.99644 

0.99619 

85 

5 

0.99619 

0.99594 

0.99567 

0.99540 

0.99511 

0.99482 

0.99452 

84 

6 

0.99152 

0.99421 

0.99390 

0.99357 

0.99324 

0.99290 

0.99255 

83 

7 

0.99255 

0.99219 

0.99182 

0.99144 

0.99106 

0.99067 

0.99027 

82 

8 

0.9P027 

0.98986 

0.98944 

0.98902 

0.98858 

0.98814 

0.98769 

81 

9 

0.98769 

0.98723 

0.98676 

0.98629 

0.98580 

0.98531 

0.98481 

80 

10 

0.98481 

0.98430 

0.98378 

0.98325 

0.98272 

6.98218 

0.98163 

79 

11 

0.98163 

0.98107 

0.98050 

0.97992 

0.97934 

0.97875 

0.97K15 

78- 

12 

0.97815 

0.97754 

0.97692 

0.97630 

0.97566 

0.97502 

0.97437 

77 

13 

0.97437 

0.97371 

0.97604 

0.97237 

0.97169 

0.97100 

0.97030 

76 

14 

0.97030 

0.96959 

0.96887 

0.96815 

0.96742 

0.96667 

0.96593 

75 

15 

0.96593 

0.96517 

0.96440 

0.96363 

0.96285 

0.96206 

0.96126 

74 

16 

0.96126 

0.96040 

0.95964 

0.95882 

0.95799 

0.95715 

0.95630 

7$ 

17 

0.95630 

0.95545 

0.95459 

0.95372 

0.95284 

0.95195 

0.95106 

72 

18 

0.95106 

0.95015 

0.94924 

0.94832 

0.94740 

0.94646 

0.94552 

71 

19 

0.94552 

0.94457 

0.94361 

0.94264 

0.94167 

0.94068 

0.93969 

70 

20 

0.93969 

0.93869 

0.93769 

0.93667 

0.93565 

0.93462 

0.93358 

69 

21 

0.93358 

093253 

0.93148 

0.93042 

0.92935 

0.92827 

0.92718 

68 

22 

0.92718 

0.92609 

0.92499 

0.92388 

0.92276 

0.92164 

0.92050 

67 

23 

0.92050 

0.91936 

0.91822 

0.91706 

0.91590 

jO.91472 

0.91355 

66 

24 

0.91355 

0.91236 

0.91116 

0.90996 

0.90875 

0.90753 

0.90631 

65 

25 

0.90631 

0.90507 

0.90383 

0.90259 

0.90133 

0.90007 

0.89879 

64 

26 

0.89879 

0.89752 

0.89623 

0.89493 

0.89863 

0.89232 

0.89101 

63 

27 

0.89101 

0.88968 

0.88835 

0.88701 

0.88566 

0.88431 

0.88295 

62 

28 

0.88295 

0.88158 

0.88020 

0.87882 

0.87743 

0.87603 

0.87462 

61 

29 

0.87462 

0.87321 

0.87178 

0.87036 

0.86892 

0.86748 

0.86603 

60 

30 

0.86603 

0.86457 

0.86310 

0.86163 

0.86015 

0.85866 

0.85717 

69 

31 

0.85717 

0.85567 

0.85416 

0.85264 

0.85112 

0.84959 

0.84805 

58 

32 

0.84805 

0.84650 

0.84495 

0.84339 

0.84182 

0.84025 

0.83867 

57 

33 

0.83867 

0.83708 

0.83549 

0.83389 

0.83228 

0.83066 

0.82904 

56 

34 

0.82904 

0.82741 

0.82577 

0.82413 

0.82248 

0.82082 

0.81915 

55 

35 

0.81915 

0.81748 

0.81580 

0.81412 

0.81242 

0.81072 

0.80902 

54 

36 

0.80902 

0.80730 

0.80558 

0.80386 

0.80212 

0.80038 

0.79864 

53 

37 

0.79864 

0.79688 

0.79512 

0.79335 

0.79158 

0.78980 

0.78801 

52 

38 

0.78801 

0.78622 

0.78442 

0.78261 

0.78079 

0.77897 

0.77715 

51 

39 

0.77715 

0.77531 

0.77347 

0.77162 

0.76977 

0.76791 

0.76604 

60 

40 

0.76604 

0.76417 

0.76229 

0.76041 

0.75851 

0.75661 

0.75471 

49 

41 

0.75171 

0.75280 

0.75088 

0.74896 

0.74703 

0.74509 

0.74314 

48 

42 

0.74314 

0.74120 

0.73924 

0.73728 

0.73351 

0.73333 

0.73135 

47 

43 

0.73185 

0.72937 

0.72737 

0.72587 

0.7*337 

0.72136 

0.71934 

46 

44 

0.71934 

0.71732 

0.71529 

0.71325 

0.71121 

0.70916 

0.70711 

45 

6O' 

6(X 

4CX  |  30' 

20' 

1O' 

0' 

f' 

SINE 

TRIGONOMETRIC  TABLES 


1 

TANGENT 

f 

0' 

10' 

20' 

30' 

40' 

5O' 

60' 

0 

0.00000 

0.00291 

.0.00582 

O.OC873 

0.01164 

0.01455 

0.01746 

89 

1 

0.01746 

0.02036 

0.02328 

0.02619 

0.02910 

0.03201 

0.03492 

88 

2 

0.03492 

0.03783 

0.04075 

0.04366 

0.04658 

0.04949 

0.05241 

87 

s 

0.05241 

0.05538 

0.05824 

0.06116 

0.06408 

0.06700 

0.06993 

86 

4 

0.06993 

0.07285 

0.07578 

0.07870 

0.08163 

0.08456 

0.08749 

85 

5 

0.08749 

0.09042 

0.09335 

0.09629 

0.09923 

0.10216 

0.10510 

84 

6 

0.10510 

0.10805 

0.11099 

0.11394 

0.11688- 

0.11983 

0.12278 

83 

7 

0.12278 

0.12574 

0.12869 

0.13165 

0.13461 

0.13758 

0.14054 

82 

8 

0.14054 

0.14351 

0.14048 

0.14945 

0.15243 

0.15540 

0.15838 

81 

9 

0.  15.838 

0.16J37 

0.16435 

0.16734 

0.17033 

0.17333 

0.17633 

80 

10 

0.17633 

0.17933 

0.18238 

0.1S534 

0.18835 

0.19136 

0.19438 

7<> 

il- 

0.19438 

0.19740 

0.20042 

0.20345 

0.20848 

0.20952 

0.21256 

78 

ls 

0.21256 

0.21560 

0.21864 

0.22169 

0.22475 

0.22781 

0.23087 

77 

13 

0.23087 

0.23393 

0.23700 

0.24008 

0.24316 

0.24G24 

0.24933 

76 

14 

0.24933 

0.25242 

0.25552 

0.25862 

0.26172 

0.26483 

0.26795 

75 

15 

0.26795 

0.27107 

0.27419 

0.27732 

0.28016 

0.28360 

0.28675 

74 

16 

0.28675 

,0.28990 

0.29305 

0.29621 

0.29938 

0.30255 

0.30578 

78 

17 

0.30573 

0.30891 

0.31210 

0.31530 

0.31850 

0.32171 

0.32492 

72 

18 

0.32492 

0.32814 

0.33136 

0.33460 

0.33783 

0.34108 

0.3443S 

71 

19 

0.34433 

0.34758 

0.35085 

0.35412 

0.35740 

0.86068 

0.36397 

70 

SO 

0.36397 

0.36727 

0.37057 

0.37388 

0.37720 

0.38053 

0.38386 

69 

21 

0.38386 

0.38721 

0.39055 

0.39391 

0.39727 

0.40065 

0.40403 

68 

22 

0.40403 

0.40741 

0.41081 

0.41421 

0.41763 

0.42105 

0.42447 

67 

23 

0.42447 

0.42791 

0.43136 

0.43481 

0.43828 

0.44175 

0.44523 

66 

24 

0.44523 

0.44872 

0.45222 

0.45578 

0.45924 

0.46277 

0.46631 

65 

25 

0.48631 

0.46985 

0.47341 

0.47698 

0.48055 

0.48414 

0.48773 

64 

26 

0.48773 

0.49134' 

0.49495 

0.49858 

0.50222 

0.50587 

0.50953 

63 

27 

0.50953 

0.51820 

0.51688 

0.52057 

0.52427 

0.52798 

0.53171 

62 

28 

0.53171 

0.53545 

0.58920 

0.54296 

0.54674 

0.55051 

0.55431 

61 

29 

0.55481 

0.55812 

0.56194 

0.56577 

0.56962 

0.57348 

0.57785 

60 

80 

0.57735 

0.58124 

0.58518 

0.58905 

0.59297 

0.59691 

0.60086 

59 

81 

0.60086 

0.60483 

0.60881 

0.61280 

0.61681 

Ot  62083 

0.62487 

58 

82 

0.62487 

0.62892 

0.68299 

0.63707 

0.64117 

0.64528 

0.64941 

57 

33 

0.64941 

0.65355 

0.65771 

0.66189 

0.66608 

0.67028 

0.67451 

56 

84 

0.67451 

0.67875 

0.68301- 

0.68728 

0.69157 

0.69588 

0.70021 

55 

85 

0.70021 

0.70455 

0.70891 

0.718S9 

0.71769 

0.72211 

0.72654 

54 

86 

0.72654 

0.73100 

0.73547 

0.73996 

0.74447 

0.74900 

0.75355 

53 

87 

0.75355 

0.75812 

0.76272 

0.76738 

0.77196 

0.77661 

0.78129 

52 

88 

0.78129 

0.78598 

0.79070 

0.79544 

0.80020 

0.80498 

0.80978 

51 

89 

0.80978 

0.81461 

0.81946 

0.82434 

0.82923 

0.83415 

0.83910 

50 

40 

0.88910 

0.84407 

0.84906 

0.85408 

0.85912 

0.86419 

0.86929 

49 

41 

0.86929 

0.87441 

0.87955 

0.88473 

0.88992 

0.89515 

0.90040 

48 

43 

0.90040 

0.90569 

0.91099 

0.91638 

0.92170 

0.92709 

0.93252 

47 

43 

0.98252 

0.98797 

0.94845 

0.94896 

0.95451 

0.96008 

0.96569 

45 

44 

0.96569 

0.97183 

0.97700 

0.98270 

0.98848 

0.99420 

1.00000 

45 

e<y 

50> 

4cx 

sex 

2O' 

icy 

O7 

COTiHGSHT 

TRIGONOMETRIC  TABLES 


COTANGENT 

O' 

10' 

20' 

3O' 

40' 

50' 

6O' 

o 

00 

343.77371 

171.88540 

114.58865 

85.93979 

68.75009 

57.28996 

89 

1 

57.28996 

49.10388 

42.96408 

38.18846 

84.86777 

81.2415828.68625 

88 

2 

28.63625 

26.43160 

24.54176 

22.903-77 

21.47040 

20.20555 

19.08114 

87 

8 

19.08114 

18.07498 

17.16934 

16.34980 

15.C0478 

14.92442 

14.30067 

86 

4 

14.3006? 

18.72674 

13.19683 

12.70621 

12.25051 

11.82617 

11.48005 

85 

5 

11.43005 

11.05943 

10.71191 

10.88540 

10.07803 

9.78817 

9.51436 

84 

6 

9.51436 

9.25530 

9.00983 

8.77689 

8.55555 

8.84496 

8.14435 

83 

7 

8.14435 

7.95302 

7.77035 

7.59575 

7.42871 

7.26873 

7.11537 

82 

8 

7.11537 

6.93823 

6.82694 

6.69116 

6.56055 

6.48484 

6.81375 

81 

9 

6.31375 

6.19703 

6,08444 

5.97576 

5.87080 

5.76987 

5.67128 

80 

10 

5.67128 

5.57638 

5.48451 

5.39552 

5.30928 

5.22566 

5.14455 

79 

11 

5.14455 

5.05584 

4.98940 

4.91516 

4.84300 

4.77286 

4.70463 

78 

12 

4.70463 

4.63825 

4.57863 

4.51071 

4.44942 

4.88969 

4.38148 

77 

18 

4.83148 

4.27471 

4.21933 

4.16530 

4.11256 

4.06107 

4.01078 

76 

14 

4.01078 

3.96165 

3.91364 

3.86671 

8.82083 

3.77595 

3.73205 

75 

15 

3.73205 

8.68909 

8.64705 

8.60588 

8.56557 

8.52609 

3.48741 

74 

16 

3.48741 

3.44951 

3.41236 

3.37594 

3.34023 

8.30521 

8.27085 

78 

17 

3.27085 

3.23714 

3.20406 

3.17159 

3.18972 

8.10842 

8.07768 

72 

18 

3.07768 

3.04749 

8.01783 

2.98869 

2.96004 

2.93189 

2.90421 

71 

19 

2.90421 

2.87700 

2.85023 

2.82391 

2.79802 

2,77254 

2.74748 

70 

20 

2.74748 

2.72281 

2.69853 

2.67462 

2.65109 

2.62791 

2.60509 

69 

21 

2.60509 

2.58261 

2.56046 

2.53865 

2.51715 

2.49597 

2.47509 

68 

22 

2.47509 

2.45451 

2.43422 

2.41421 

2.39449 

2.37504 

2.35585 

67 

23 

2.35585 

2.33693 

2.81826 

2.29984 

2.28167 

2.26374 

2.24604 

66 

24 

2.24604 

2.2285? 

2.21182 

2.19430 

2.17749 

2.16090 

2.14451 

65 

25 

2.14451 

2.12832 

2.11233 

2.09654 

2.08094 

2.06553 

2.05030 

64 

28 

2.05030 

2.03525 

2.02039 

2.00569 

1.99116 

1.97680 

1.96261 

68 

87 

.98261 

1.94858 

1.93470 

1.92098 

1.90741 

1.89400 

1.88078 

62 

28 

.88073 

1.86760 

1.85462 

1.84177 

1.82907 

1.81649 

1.80405 

61 

29 

.80405 

1.79174 

1.77955 

1.76749 

1.75556 

1.74375 

1.73205 

60 

80 

.73205 

1.72047 

1.70901 

1.69766 

1.68643 

1.67580 

1.66428 

59 

SI 

.66428 

1.65337 

1.64256 

i.63i  as 

1.62125 

1.61074 

1.60033 

58 

82 

.60033 

1.59002 

1.57981 

1.56969 

1.55966 

1.54972 

1.58987 

57 

83 

.53987 

1.53010 

1.52043 

1.51084 

1.50138 

1.49190 

1.48256 

50 

34 

.48256 

1.47330 

1.46411 

1.45501 

1.44598 

1.48703 

1.42815 

55 

35 

1.42815 

1  41934 

1.41061 

1.40195 

1.39336 

.88484 

1.87638 

54 

86 

1.87638 

1.36800 

1.35968 

1.35142 

1.84323 

.88511 

1.82704 

58 

37 

1.32704 

1.31904 

1.31110 

1.30323 

1.29541 

.28764 

.27994 

52 

38 

1.27994 

1.27230 

1.23471 

1.25717 

1.24969 

.24227 

.23490 

51 

39 

1.23490 

1.22758 

1.22031 

1.21310 

1.20593 

.19882 

.19175 

50 

40 

1.19175 

1.18474 

.17777 

1.17085 

1.16398 

.15715 

.15037 

49 

41 

1.15037 

1.14368 

.13694 

1.13029 

1.12369 

.11713 

.11061 

48 

42 

1.11081 

1.10414 

.09770 

1.09181 

1.08496 

1.07864 

.07237 

47 

43 

1.07237 

1.06613 

.05994 

1.05378 

1.04766 

1.04158 

1.08553 

46 

44 

1.03553 

1.02952 

.02355 

1.01761 

1.01170 

1.00583 

1.00000 

45 

60' 

50' 

40' 

3O' 

2<y   icy 

O/ 

T4NGSNT 

TRIGONOMETRIC  TABLES 


2 
£ 

SECAMS 

H 
I 

0' 

10' 

20' 

3O' 

40' 

6CK 

6O' 

0 

1.00000 

1.00001 

1.00002 

1.00004 

1.00007 

1.00011 

1.00015 

89 

1 

1.0C015 

1.00021 

1.00027 

1.00034 

1.00042 

1.00051 

1.00061 

88 

a 

1.00061 

1.00072 

1.00083 

1.00095 

1.00108 

1.00122 

1.00137 

87 

3 

1.00137 

.00153 

1.00169 

1.00187 

1.00205 

1.00224 

1.00244 

86 

4 

1.00244 

.00265 

1.00287 

1.00309 

1.00333 

1.00357 

1.00382 

85 

5 

1.00382 

.00408 

1.00435 

1.00463 

1.00491 

1.00521 

1.00551 

84 

6 

1.00551 

.00582 

.00(514 

1.00647 

1.006S1 

1.00715 

1.00751 

83 

7 

1.00751 

.00787 

.00825 

1.00863 

1.00W2 

1.00942 

1.00983 

82 

8 

1.00983 

.01024 

.01067 

1.01111 

1.01155 

1.01200 

1.01247 

81 

9 

1.01247 

.01294 

.01342 

1.01391 

1.01440 

1.01491 

1.01543 

80 

10 

1.01543 

1.01595 

.01649 

1.01703 

1.01758 

1.01815 

1.01872 

79 

11 

1.01872 

1.00930 

.01989 

1.02049 

1.02110 

1.02171 

1.02234 

78 

12 

1.02284 

1.02298 

.02362 

1.02428 

1.02494 

1  .02562 

1.02630 

77 

13 

1.02630 

1.02700 

.02770 

1.02842 

1.02914 

1.02987 

1.03061 

76 

14 

1.03061 

1.03137 

1.03213 

1.03290 

1.03368 

1.03447 

1.03528 

75 

15 

1.03528 

1.03609 

1.03691 

1.03774 

1.03858 

1.03944 

1.04080 

74 

16 

1.04030 

1.04117 

1.04206 

1.04295 

1.04385 

1.04477 

1.04569 

73 

17 

1.04569 

1.04663 

1.04757 

1.04853 

1.04950 

1.05047 

1.05146 

72 

18 

1.05146 

1.05246 

1.05347 

1.05449 

1.05552 

1.05657 

1.05762 

71 

19 

1.05762 

1.05869 

1.05976 

1.06085 

1.06195 

1.06306 

1.06418 

70 

20 

1.06418 

1,06531 

1.06645 

1.06761 

1.06878 

1.06995 

1.07115 

69 

21 

1.07115 

1.07235 

1.07356 

1.07479 

1.07602 

1.07727 

1.07853 

68 

22 

1.07858 

1.07981 

1.08109 

1.08239 

1.08370 

1.08503 

1.08636 

67 

23 

1.08686 

1.03771 

.08907 

1.09044 

1.09183 

1.06323 

1.09464 

66 

24, 

1.09464 

1.09606 

.09750 

1.09895 

1.10041 

1.10189 

1.10338 

65 

25 

1.10338 

1.10488 

.10640 

1.10793 

1.10947 

1.11103 

1.11260 

64 

2a 

1.11260 

1.11419 

.11579 

1.11740 

1.11903 

1.12067 

1.12283 

63 

27 

1.12233 

.12400 

.12568 

1.12738 

1.12910 

1.13083 

1.13257 

62 

23 

1.13257 

.13433 

.13610 

1.13789 

1:13970 

1.14152 

1.14385 

61 

29 

1.14335 

.14521 

1.14707 

1.14896 

1.15085 

1.15277 

1.15470 

60 

30 

1.15470 

.15665 

1.15861 

1.16059 

1.16259 

1.16460 

1.16668 

59 

31 

1.16663 

.16888 

1.17075 

1.17283 

1.17498 

1.17704 

1.17918 

58 

82 

1.17918 

.18133 

1.18350 

1.18589 

1.18790 

1.19012 

1.19286 

57 

33 

1.19236 

.19463 

1.19891 

1.19920 

1.20152 

1.20386 

1.20622 

56 

54 

1.20622 

.20859 

1.21099 

1.21341 

1.21584 

1.21830 

1.22077 

55 

35 

1,22077 

1.22327 

1.32579 

1.22883 

1.23089 

1.23347 

1.23607 

54 

36 

1.23607 

1.23889 

1.24134 

1.24400 

1.24669 

1.24940 

1.25214 

53 

37 

1.25214 

1.25489 

1.25767 

1.26047 

1.26330 

1.26615 

1.26902 

52 

38 

1.26902 

1.27191 

1.27463 

1.27778 

1.28075 

1.28374 

1.28676 

51 

39 

1.28676 

1.28980 

1.29387 

1.29597 

1.29909 

1.30223 

1.80541 

50 

40 

1.80541 

1.3*3861 

1*.  81183 

1.81509 

1.31837 

1.82168 

1.82501 

49 

41 

1.32501 

1.32838 

1.33177 

1.83519 

1.33864 

1.34212 

1.84563 

48 

42 

1.34563 

1.84917 

1.35274 

1.85384 

1.85997 

1.36363 

1.36733 

47 

43 

1.36733 

1.87105 

1.37481 

1.87860 

1.88242 

1.38628 

1.89016 

46 

44 

1.89016 

1.89409 

1.89804 

1.40203 

1*40608 

1.41012 

1.41421 

45 

6O' 

6O' 

4O' 

SO' 

2O* 

10' 

O 

1 

COSECANTS 

S 

TRIGONOMETRIC  TABLES 


-7-   •',:..  iJOSECMW  ; 

O' 

1O' 

20' 

30' 

40> 

50' 

60' 

0 

00 

343.77516 

171.88831 

114.59301 

85.94561 

68.75736 

57.29869 

89 

1 

57.29869 

49.11406 

42.97571 

38.20155 

34.38232 

31.25758 

28.65371 

88 

2 

28.65371 

26.45051 

24.56212 

22.92559 

21.49368 

20.23028 

19.10732 

87 

3 

19.10732 

18.10262 

17.19843 

16.38041 

15.63679 

14.95788 

14.33559 

86 

4 

14  .-33559 

13.76312 

13.23472 

12.74550 

12.29125 

11.86837 

11.47371 

85 

5 

11.47371 

31.10455 

•10.75849 

10.43343 

10.  12752 

9.83912 

9.56677 

84 

6 

9.56677 

9.30917 

9.06515 

8.83367 

8.61379 

8.46466 

8.20551 

83 

7 

8.20551 

8.01565 

7.83443 

7.66130 

7.49571 

7.33719 

7.18530 

82 

8 

7.18530 

7.03962 

6.89979 

6.76547 

6.63633 

6.51208 

6.39245 

81 

9 

6.39245 

6.27719 

6.16607 

6.05886 

,  5.95536 

5.85539 

5.75877 

80 

10 

5.75877 

5.66533 

5.57493 

5.48740 

5.40263 

5.32049 

5.24084 

79 

11 

5.24084 

5.16359 

5.08863 

5.01585 

4.94517 

4.87649 

4.80973 

78 

12 

4.80973 

4.74482 

4.68167 

4.62023 

4.56041 

4.50216 

4.44541 

77 

13 

4.44541 

4.39012 

4.33622 

4.28366 

4.23239 

4.18238 

4.13357 

76 

14 

4.13357 

4.08591 

4.03938 

3.99393 

3.94952 

3.90613 

3.86370 

75 

15 

3.86370 

8.82223 

3.78166 

8.74198 

8.70315 

3.66515 

3.62796 

74 

16 

8.62796 

3.59154 

3.55587 

3.52094 

3.48671 

3.45317 

3.42030 

73 

17 

3.42030 

3.38808 

3.35649 

3.32551 

3.29512 

3.26531 

3.23607 

72 

18 

3.23607 

3.20737 

3.17920 

3.15155 

3.12440 

3.09774 

8.07155 

71 

19 

3.07155 

3.04584 

3.02057 

2.99574 

2  97135 

2.94737 

2.92380 

70 

20 

2.92380 

2.90063 

2.87785 

2.85545 

2.83342 

2.81175 

2.79043 

69 

21 

2.79043 

2.76945 

2.74881 

2.72850 

2.70851 

2.68884 

2.66947 

68 

•22 

2.66947 

2.65040 

2.63162 

2.61313 

2.59491 

2.57698 

2.55930 

67 

23 

2.55930 

2.54190 

2.52474 

2.50784 

2.49119 

2.47477 

2.45859 

6ft 

24 

2.45859 

2.44264 

2.42692 

2.41142 

2.39614 

2.38107 

2.36620 

65 

25 

2.36620 

2.35154 

2.33708 

2.32282 

2.30875 

2.29487 

2.28117 

64 

26 

2.28117 

2.26766 

2.25432 

2.24116 

2.22817 

2.21535 

2.20269 

63 

27 

2.20269 

2.19019 

2.17786 

2.16568 

2.15366 

2.14178 

2.13005 

62 

28 

2.13005 

2.11847 

2.10704 

2.09574 

2.08458 

2.07356 

2.06267 

61 

29 

2.05267 

2.05191 

2.04128 

2.03077 

2.02039 

2.01014 

2.00000 

60 

30 

2.00000 

1.98998 

1.98008 

1.97029 

1.96062 

1.95106 

1.94160 

59 

31 

.94160 

1.93226 

1.92302 

1.91388 

1.90485 

1.89591 

1.88709 

58 

32 

.88708 

1.87834 

1.86990 

1.86116 

1.85271 

1.84435 

1.83608 

57 

33 

.83608 

1.82790 

1.81981 

1.81180 

1.80388 

1.79604 

1.78829 

56 

34 

.78829 

1.78082 

1.77303 

1  76552 

1.75808 

1.75073 

1.74345 

55 

35 

.74345 

1.73624 

1.72911 

1.V2205 

1.71506 

1.70815 

1.70130 

54 

38 

.70130 

1.69452 

1.68782 

1.68117 

1.67460 

1.66809 

1.66164 

53 

37 

.66164 

1.65526 

1.64894 

1.64268 

1.03648 

1.63035 

1.62427 

52 

38 

.62427 

1.61825 

1.61229 

1.60839 

1.60054 

1.59475 

1.58902 

51 

39 

.58902 

1.58333 

1.57771 

1.57213 

1.56661 

1.56114 

1.55572 

50 

40 

.55572 

1.55036 

1.54504 

1.53977 

1.53455 

1.52938 

1.52425 

40 

41 

.52425 

1.51918 

1.51415 

1.50916 

1.50422 

1.49933 

1.49448 

48 

42 

.49448 

1.48967 

1.48491 

1.48019 

1.47551 

1.47087 

1.46628 

4? 

43 

.46628 

1.46173 

1.45721 

1.45274 

1.44^31 

1.44391 

1.43956 

4'j 

44 

1.43956 

1.43524 

1.43096 

1.42672 

1.42251 

1.41835 

1.41421 

45 

60' 

6O' 

40' 

SO' 

20' 

1O' 

O/ 

1 
g) 

SECANTS 

« 

YA  0686 


**s  > 


